An example of some changes in one episode:
"The Corbomite Maneuver" Screenshots and Video -- Side by side comparison of Remastered and Original
More obvious are the changes to these two episodes (although the site does not point out what I thought was the most impressive change, the visual effects of the ground phaser cannon):
“The Menagerie” (Part 1) Screenshots and FX Video [Updated]
“The Menagerie” (Part 2) Screenshots and FX Video
In watching the remastered version of "The Enterprise Incident," someone familiar with the original version will notice that the three Romulan vessels which surround the Enterprise have been altered. For the most part, this has been done in relatively minor ways (such as adding the bird of prey logo seen on the bottom of the T'Liss in "Balance of Terror" to the bottom of the D7 ships). However, one very noticeable change is the removal of one of the D7 ships entirely, which has been replaced by a T'Liss. In an effort to give an explanation of this within the context of the Star Trek universe itself, as well as to explain away some inconsistencies in the various novels which include the Romulan Commander from "The Enterprise Incident," I've written a short fan fiction tale, which you can find here:
Gia'na t'Prell, Romulan Temporal Sciences -- The Prize, and the Price (Fan Fic).
For those who may not think this is a "good" explanation, bear in mind that, according to the TOS episode "Space Seed," something occurred in the 1990s, which was not evident when a vessel from the 24th century went back into 1996:
Both "Space Seed" and Star Trek II: The Wrath of Khan gave the dating of the Eugenics Wars as the 1990s. "Doctor Bashir, I Presume", set in 2373, however referenced the Wars as having occurred two centuries prior to the episode, placing the Wars in the late 22nd century. Episode writer Ronald D. Moore later admitted that this dialogue statement was a production error, a line he had taken from The Wrath of Khan, but accidentally forgot to account for the episode being a century later. (AOL chat, 1997) The original dating of the Wars was reaffirmed when Phlox stated in "Borderland" that Arik Soong's Augments were pretty sophisticated for 20th century genetics. Phlox later mentions to the Klingons that genetic engineering on Earth was "banned decades ago," suggesting that the ban was not necessarily adopted by Humans immediately after the Eugenics Wars.
In "Space Seed", Spock describes the Eugenics Wars as "the era of your last so-called world war," suggesting this conflict could be World War III. In TOS: "Bread and Circuses", it is stated by Spock that 37 million died in World War III – consistent with Phlox's assertion that over 30 million died in the Eugenics Wars (again connecting World War III and the Eugenics Wars) – but not Riker's claim that 600 million died in the nuclear conflict in the 2050s. As Spock was speaking in the context of despotism, and what constitutes despotic "responsibility" is open to interpretation, his statement may not give the total death count.
In TNG: "Up The Long Ladder", Data stated that Humans were still recovering from the effects of World War III in the early 22nd century. This statement makes more sense within the context of a mid 21st century war than that of a late 20th century war, suggesting that World War III and the Eugenics Wars are not the same conflict, as confirmed in Star Trek: First Contact.
In a two-part episode of Star Trek: Voyager ("Future's End" and "Future's End, Part II"), the crew of the Voyager goes back in time to the 1990s, and there is no evidence of any "Eugenics Wars." No "official" explanation has been given for this, but we have this information:
Although this two-part story is mostly set in 1996, there is no allusion made to the Eugenics Wars which, according to both TOS: "Space Seed" and Star Trek II: The Wrath of Khan, took place at this time. Prior to this episode's first airing, Jeri Taylor told a convention audience, "I think that those of us who entered into the Nineties realize the Eugenics Wars simply aren't happening and we [the writers] chose not to falsify our present, which is a very weird thing to do to be true to it." (Star Trek Monthly issue 22) Furthermore, in an audio commentary for Star Trek: First Contact, Brannon Braga states that it was decided not to have the Eugenics Wars in this episode because "it would just be kind of strange." This decision was also made, however, because Voyager's writing staff didn't want to bog the "Future's End" two-parter down by having to explain the Eugenics Wars to the majority of the audience (who, according to the series' research, were irregular viewers of Voyager and not hard-core fans of the series). The DS9 episode "Doctor Bashir, I Presume" (produced soon after this one) mentions the wars as having taken place in the 22nd century and not the 20th century, which may account for the wars' exclusion from this episode's two-parter (although writer Ronald D. Moore himself admitted that the DS9 episode's dating of the wars was merely an error on his part–recalling the already iffy "two centuries" quote from "Space Seed" and then forgetting that the DS9 episode took place 106 years later–despite Joe Menosky suspecting differently). (Cinefantastique, Vol. 29, No. 6/7, p. 110) The novel series Star Trek: The Eugenics Wars would later seek to retcon them as a secret history in which various Augments, largely fighting amongst one another, were responsible for numerous real-life calamities from the early 1990s, making seemingly isolated events all part of one wider conflict; ironically, Los Angeles, the city whose untouched-by-war appearance brought their existence into question, is actually portrayed as an EW "battlefront", its 1992 race riots being one such incident.
Some fans, however, have speculated that the backstory (shown in the beginning of the first of the two episodes) for the two-part "Future's End" story is itself the reason for the omission; namely, that the timeline became altered as a result of the actions of Henry Starling. Certainly the episode suggests the possibility when it refers to the microcomputer revolution of the 20th century as the result of time travel and the exploitation of future technology by Starling.
It would not be much challenge for a ship with time travel capacity to arrange for one ship from the past to be reassigned or delayed or the like, and replaced by another (or even for the time-traveling vessel itself to take the place of the original vessel, particularly with the aid of photonic technology which could disguise the time-traveling vessel), but enough of this, lest I give spoilers to the short story I have written).