Does the GOG version already come with the 64-bit Upgrade Patch and the 64-bit Exclusive Content Patch?
http://pcgamingwiki.com/wiki/Far_Cry

You can't use the 64-bit patches with GOG's version. The 64-bit patches transformed the version 1.3.2 into a native 64-bit application. Later on, Crytek released further patches with additional features, but only for the 32-bit version, so the newest version is 1.4 (32-bit). That's the version GOG is offering. There's no 64-bit version of 1.4.

Is the 64-bit version of Far Cry really necessary? I played GOG's version of Far Cry on Win7 64-bit and it ran fine. Plus it looks like the "Far Cry AMD 64 Exclusive Content Update Patch" only works with AMD chips according to GamersHell.com.

No, it was more of a promo cooperation between Crytek and AMD to promote the game and 64-bit technology in 2004. Native 64-bit support gave the game a boost under Windows XP 64-bit (with AMD processors). The 32-bit version is running fine on modern machines, no matter if you're using 32 or 64-bit Windows. Though you're missing some (but not much) exclusive bonus content that's included in the patch.

What exactly was the other content for the patch? Another level? The GamersHell.com site didn't mention much information.

Very little, actually. Mostly it was minor improvements to the graphics, such as longer view distance.

Afaik it's this:

All Levels:
- Improved terrain textures
- Increased view distance
- Offset bump mapping added for rock and stone objects
- More insects and birds

On the Pier Level:
- New beach road with additional vehicle
- Barrel storage camp
- Opened more space to explore

Pier and Boat:
- New terrain textures with shader (next-gen technology)

Two multiplayer levels:
- Stronghold
- Gorge

WOW, thank you all for the helpful answers. I'm going to mark the first helpful reply as the solution.