As comparing the original and remake to identify what was in both games would serve difficult, I am only including an overarching view of new or changed story/gameplay content that only exists in the remake.
The content below references this guide, as well as information that stems from my playthrough of the original FF7 many many years ago. These are likely not all of the changes, but it's a good start.
The members of Avalanche - Jessie, Wedge, and Biggs - have much more screen time in the remake than the original. Chapter 4 consists of a lot of original content surrounding these members.
Jessie, Wedge, and Biggs all originally died in the Sector 7 collapse. At the end of the Remake, Biggs appears to have survived. Did Jessie and Wedge survive as well? That remains to be seen
Chapter 4 introduces an original character Roche (SOLDIER 3rd Class)
Chapter 3 introduces an original character, Chadley (Shinra research intern).
Chapter 9 introduces original characters, Andrea Rhodea, Madam M, Chocobo Sam, and Leslie Kyle.
The mysterious Whispers that follow Aerith around the entire game are brand new.
Sephiroth makes his appearance much earlier than in the original
In the final sequence of the game's conclusion, we get a small glimpse of Zack Fair. In the original game, Zack's cutscene is optional and found much later in the story. If you played the original or Crisis Core, Zack is known to have die in battle protecting Cloud, but we see a different take of these events and hints towards a possible survival.
- Chadley introduces new "research" activities to obtain various types of materia, including summons
- Wall Market introduces new mini-games, such as dancing, pull-ups, and squatting
- Wall Market introduces Corneo's Colosseum, which consists of battle challenges and rewards
- Various towns include "odd-jobs"(or side quests)