Rescue Team DX faithfully keeps the story and dungeons of 2006, but modernizes the game a great deal. Here are the new features, based on Poképédia.

Magma Cavern.
New featureWhat it changes
Auto modeThe L button moves the team until an enemy approaches
Best attackThe A button picks the most effective move against the opponent
Bigger teamsUp to five extra recruits in a dungeon, eight members in total
Switching leadersPossible in every dungeon, even during the story, with the + button
New PokémonMostly evolutions and pre-evolutions from the first three generations that appeared later, such as Probopass or Budew
Modern movesMoves up to the seventh generation; Aero Blades and Reaper are gone
Powerful PokémonOne per dungeon, sometimes shiny, recruitable; when defeated, it leaves a chest
Mystery HousesOpened with an Invitation Letter, they contain items and Pokémon that join the team

To compare with the original, see the differences between Red and Blue.