The strange thing is that I didn't notice these differences while playing the Remake. Sure, some things seemed more fleshed out, but I was certain that I kept seeing echoes of people and events from the original that I now know didn't exist at all. More surprisingly, there were a few characters and story elements that I thought were made up to flesh out the Remake, but were actually in the original. I'm going to go into detail for a few of the changes, so if you don't want spoilers, you might want to stop now. However, there is no way that I can give a comprehensive list; I'm just going to mention a few highlights.
Many of the changes are just in the overall size and scope of the game. The sewers are a maze of interconnecting passages in the Remake, but only two small screens in the original. The same is true for the destroyed area between the slums in sectors 5 and 7; in this case, it's only one screen in the original. In the remake, there are mechanical hand puzzles which I could have sworn were from the original, but the original only has a static robot hand in the background of one of the scenes. Oddly, Shinra tower--while it still has the conference room scene and the choice between elevator or stairs--has entirely different areas when comparing the remake to the original.
The biggest change in the Remake is the inclusion of the Whispers, Dementor-looking ghosts that interact with the main characters from time to time. They usually show up around the iconic moments from the first game to make sure destiny stays on course, but they also appear when events start to diverge. They stay to make certain that some characters die, but also save some characters (one in particular) from an unexpected death.
The greatest surprise for me was that Cloud does not get to see the Avalanche hideout in the Remake. I thought that I somehow missed something or made a mistake when I was playing. However, Cloud in the Remake is not as much a part of Avalanche as he was in the original. It's clear that Barret really doesn't trust him, and so Cloud is not only not invited into the hideout, but he isn't even invited on the second reactor mission. Here is where the Whispers come in and injure one of the other Avalanche members to ensure Cloud's participation. It was an interesting tonal shift between the games, but it actually made the shift of animosity to grudging trust to eventual friendship between Cloud and Barret make a great deal more sense.
I was also surprised to find Johnny in the original game. He's an annoying, but often amusing, pest in the Remake that I thought was added in like many of the other characters, yet there he was in the original, standing outside the Honey Bee. Which brings me to another change: I thought I remembered seeing the inside of the Honey Bee in the original; I was wrong, but some lines said inside it in the Remake are said outside in the original.
Wow! I could really go on: the addition of Jesse's family, the consistency with Aerith's mother's story and house, Cloud's new flashbacks, Cloud's consistency with some flashbacks, the difference in Red XIII's introduction and Hojo's lab, the new role of Midgar's mayor, Shinra's plot to implicate a Wutai connection to Avalanche ... and so much more. I am amazed at what they did with the connections and yet the subtle (and not so subtle) differences. In the end, I feel that the changes make for a better story, one that is closer to my memory of the original than the original itself actually is.