The Enterprise Refit of 2271
There is no canon evidence in any episode, but according to the Star Trek Encyclopedia the original starship Enterprise NCC-1701 was launched in 2245 (discounting the admiral's faulty claim in "Star Trek III" that the ship was just 20 years old in 2284, which can't be true). The Enterprise was apparently first refurbished some time between 2254 and 2265. There are modifications to the deflector dish, the side walls of the engineering hull, the nacelle spires, the nacelle aft ends and the bridge module from the first pilot to the series version of the studio miniature. Also, while the crew complement was said to be 200 under Captain Pike ("The Cage"), it increased to 400 crew members under Captain Kirk in the first season of TOS. It is a minor point that some instruments on the main bridge and the colors of the bridge were changed too during this time, for this can be accomplished in very brief time and doesn't necessarily point to an overall refit. As the change of its exterior insinuates, it seems to be a completely new bridge module anyway.
Side note Also, the Enterprise engineering room was modified some time in the second TOS season and a dilithium chamber was added, among a couple of other changes. Since the overall dimensions remained almost identical, it was definitely supposed to be still the very same room. It may have been the original intention that engineering was located in the saucer, just ahead of the impulse engines, thus the red glowing tubes in the background. However, plausibility requires the engine room to be inside the engineering hull and closer to the warp pylons, considering that the dilithium chamber is an essential part of the warp drive.
While the refit cycle established for the Galaxy class in the STTNG Technical Manual is 20 years, it seems to have been less than 15 years in the case of the Constitution class and original Enterprise. A very extensive refit of the ship took place from 2269, when Kirk's five-year mission had just ended, to 2271 ("Star Trek I"). Although the overall appearance of the Enterprise remained roughly the same, an examination of the components reveals that very little can have been left of the old ship. The question arises if it is justified at all to call it still the same ship.
Note All the following dimensions were measured on 1/1200 scale diagrams of the two ships, assuming an overall length of 289m for the original version and 305m for the refit. There might be insignificant errors in my measurements as well as in the schematics.
Comparison of Original and Refit
Most obviously, the old cylindrical nacelles (length: 156m, diameter: between 13.4m and 16.2m) were replaced with new angular ones (length: 152m, width: 12m, height: 15.6m). Since rather the propulsion technology than only the shell of the engines would be subject to change, we may safely conclude that the warp coils and everything else inside the nacelles are new as well. It is questionable anyway whether the old components would still fit into the new (smaller) housing. The new glowing warp field grille (visible on the inward sides) is another sign that there are new warp coils. The most obvious evidence, however, is that the Miranda class seems to employ exactly the same nacelle type. No matter if the Miranda was developed before or after the Constitution refit, the nacelle looks like it is a new standard type and not a modification of the original Constitution nacelle.
The warp nacelle struts are completely new too, and they are connected to the engineering hull much closer to the neck than before.
The secondary hull was overall shifted backward (relative to the saucer) and only roughly retained its original shape. While the old one was essentially a conical cylinder (actually without the curvature at its bottom that is visible in some drawn side views), the new one is more rounded, like a wooden vat. The length of the engineering hull has increased from 99m to 111m, its maximum diameter by 4.6m. Interestingly the new, evidently "fatter" hull doesn't look like it is even longer than the old one. The true proportions are deceptive because the deflector dish was sticking out from the old hull while the forward end is blunt on the new version. If we disregard the separate deflector dish on the old Enterprise, the two engineering hull forward ends are largely congruent, while it looks like just the shuttlebay has been extended to the rear, with a new, more sloped sliding door. Nevertheless, almost the complete outer surface must be new, since the two versions have hardly any hull plates with corresponding shapes. Thus, it is plausible that the windows are all new as well.
Pertaining to the interior of the secondary hull, the most obvious changes are that the shuttlebay was significantly enlarged (not on the later Enterprise-A where the shuttlebay as it is visible in "Star Trek V" looks like it is much smaller than even in TOS) and the ship now has a M/ARA occupying multiple decks. It is possible that the former engine control room did not include the actual warp core and thus looked very different. Yet, the dilithium chamber of the warp core or at least its load lock was located there at latest since 2268 (as seen in TOS: "Elaan of Troyius"). So it is obvious that the old engine room doesn't exist any longer. Moreover, since the new warp pylons are sloped instead of running straight up from the hull, we have to assume that the path of the power transfer conduits inside the engineering hull has accordingly changed, and so would have the location of the warp core and main engineering. The absence of a external deflector dish would necessitate moving the whole sensor and other equipment backward.
Summarizing, the engineering hull could easily be a completely new structure, rather than a heavily modified part of the original Enterprise. Maybe something of the old structural frame is left, but it would be rather little of it, considering that the whole surface shape is different and the interior partition has noticeably changed too.
The connecting dorsal was not only modified by installing the double photon torpedo launcher with its rectangular housing. The new neck has about the same width (in the front view), and a possible small deviation in the range of less than 1m could easily be explained with either thicker or thinner hull layers on the new neck. All other dimensions, on the other hand, were significantly changed. The old neck was merely 24m long (in the side view) at its base at the engineering hull and 36m at the saucer bottom, while the new one measures 42m at both ends. Taking the distance between the centerline of the engineering hull and the underside at the saucer rim, the new neck is also 3.5m or one deck taller. The window locations, maybe even the deck heights, have been altered as well. It is obvious that not just the surface of the saucer, neck and engineering hull was renewed, but that these components were actually dis- and re-assembled. Like the engineering hull, the neck is either completely or mostly new.
Note Four windows in the neck that are vertically lined up along the turbolift shaft are congruent on the two neck versions. But since the surface is likely new, this may be regarded as a coincidence.
The old saucer diameter was 126m, and most obviously it was extended to 142m. The two decks in the new 8m thick ring around the saucer might contain another corridor and/or a turbolift shaft and numerous staterooms, for instance. The rest of the surface remained largely the same. At least, we may presume that the structural frame underneath the hull plates was not changed. In particular, the central portion of the saucer, including the lower sensor dome, is exactly congruent if we presuppose 289m length for the original version and 305m for the refit. The saucer thickness (the height of the two large decks) remains the same as well. Actually, this perfect correspondence is the most important evidence for the relative sizes of all other components, whose proportions were more or less radically changed.
The details on the saucer surface, on the other hand, including the deflector grid, phaser emitters and windows are entirely new. The same applies to the sensor dome and the impulse engines. Unlike the neck, the saucer is big enough to allow the interpretation that its interior partition remained largely unchanged though. Even the big recreation deck visible in "Star Trek I" could always have been there and was just not shown in TOS. Nevertheless, it is undeniable that the internal equipment like wall, floor and ceiling panels and doors is new, owing to technical as well as "stylistic advancements", so that merely the ground plan may have been retained.
Since starship bridges are supposed to be interchangeable modules anyway, it is easy to explain that the new Enterprise has a new bridge module, again two decks tall, but now with two turbolift shafts. The ground plans of the two bridge modules are a bit different but this is rather a minor problem considering the many other modifications.
Regarding vessels with a long lifespan, whose later refits looked only vaguely like the original version and more like newer classes, we can refer to the Midway class of the US Navy as a real-life example. The aircraft carrier USS Midway CV-41 (ex-CVA-41, ex-CVB-41) was commissioned in 1945 and went through a number of refits until she was retired in 1992. Among other modifications, the layout of the flight deck was radically changed and its area increased from the original to the final appearance of the ship. The later flight decks and many other parts of the ship arguably looked much more like those of the Forrestal CV-59 and the following supercarriers than like those of the Essex class, the vessels serving in WWII from which the Midway class was derived. The gradual refit of the Midway is not exactly the same that happened with the Starship Enterprise, but the final result is pretty much equivalent. Maybe the Enterprise just had overall longer overhaul cycles, and all the recent achievements were slated to be incorporated at once when the ship was literally taken apart around 2269.
An example of a vessel that was visually modified even more extensively and in one step is the USS Albany CA-123 that was converted from a heavy cruiser to a guided-missile cruiser from 1958 to 1962. The deck superstructures of the refit has nothing in common with the original version. The Albany (previously Oregon City class) became the lead ship of the new Albany class, and perhaps in the same vein the Enterprise (previously Constitution class) was the first ship of the Enterprise class, as a sign in the simulator room in "Star Trek II" insinuates.
Summarizing, only the internal structure of the saucer and a bit of the engineering hull and neck of the original USS Enterprise can possibly have survived the upgrade in 2271. Hence, it would be easier to explain the considerable differences in that the Enterprise "refit" is actually a new ship of a new class. However, both versions are unequivocally called "Constitution class" (only the TOS dedication plaque says "Starship Class") and the movie dialogues (at least Kirk's lines) imply that this is still "his" Enterprise - with the same name and registry. Matt Jefferies's original version of 2265 and Andrew Probert's new Enterprise of 1978 are both well-designed and very detailed and were prominently featured in the series and the movies, respectively. This is why the countless differences are striking and can't be simply explained by inaccurate presentation on screen, as it might be possible with alien ships. Therefore it seems necessary to accept that Starfleet built something very close to a new ship, but retained several parts of the old ship - maybe to save costs. This may have sufficiently justified to keep up the individual ship names of the venerable Constitution class (while the new class may be referred to as "Enterprise class").
Starship Gallery - Constitution - original and refit
Starships in TOS and TOS Remastered - "before & after" comparison of all ships that appeared in the series
The Enterprise Legacy - a compilation about the eight starships to bear this name
Thanks to Andrew Loates for throwing in the USS Midway as a possible real-life equivalent and to James Yeh who pointed me to the USS Albany.
Ship of Theseus @ Wikipedia