What are the differences between Baldur's Gate Enhanced edition and the classic one?

The new companions correspond to some of the new class choices that are available. The Blackguard kit for Paladin, the Monk (base class, but implemented interally as a Thief variant), and Wild Mage kit for the Wizard. There are also the Sorceror and Barbarian classes, and a few other kits that BG1 didn't have.

There are some changes to the user-interface to scale it to larger resolutions, and mouse-wheel or pinch can zoom in or out the scaling.

Some changes that will also be evident from comparing Baldur's Gate 2 to Baldur's Gate, because they are doing the same as the Tutu mods and running a fixed-up version of the Baldur's Gate 2 version of the Infinity Engine to run the original game: two items of note here, aside from the above

• New races
• Dual-wielding

I believe the team wanted to implement more new races but weren't allowed to by Atari or Wizards (not sure whom), but you get Half-Orc addition at least.

Various mods out there fixed issues with the original game. A short list from this forum post explains which ones are no longer needed:

• ToBEx fixes game engine bugs by modifying the executable: To be honest, I'm having trouble weeding out what here is just hacks/mods and what's actually been fixed. I believe a scroll-bar on the kit selection UI means you can add more kits to the game without not being able to select the last added ones (something modders had to deal with by removing other kits).
• BG1 Fixpack
• BG2 Fixpack: Fixes to descriptions, sounds, barbarian/berserker rage/enrage, and a ton more.
• Baldurdash fixes for BG1
• Baldurdash fixes for BG2, where it applies to the engine/rules/etc and not BG2 specific content
• Dudleyville fixed over 700 things. Some of which are fixes to errors in NPCs (illegal proficiencies for classes, for example), spells
• BGT or its fixes
• Widescreen mod
• One Pixel Productions list of changes are:
  • BG1 Paperdoll and inventory item ports, except for potions.
  • BG1 Potion Graphics
  • BG1-style Flaming Swords
  • Flaming Short Swords (with item and mod item patches)
  • New Paperdolls (human, half-orc, elven, halfling, dwarf)
  • Legacy Shields - Bucklers (D1)
  • Legacy Shields - Small Shields (D2)
  • Legacy Shields - Medium Shields (D3)
  • Colourable Quarterstaves (with item patches)

Edit: I just realized there's one actual removal-- there is no longer EAX support for position audio and reverb. It's a little disappointing but not a deal-breaker, IMO.