I'm sure someone would be able to expand upon this further, but from what I understand there were four things:
- Different case and physical layout (e.g. controllers have storage slots on the sides of the main unit).
- Cartridge slot has fewer pins as it didn't support the "10NES" lockout chip.
- One of the controllers had a microphone on it for voice activated gaming.
- The expansion port on the bottom of the NES versus the DB15 port on the Famicom.
The first one is obvious cosmetically.
For #2, all the signals of the Famicom cartridge slot were available on the NES and as such there were adapters available which would allow you to use a Famicom cartridge on a NES.
The third one is the most intriguing since it is one feature which didn't appear at all on the NES. There were only a few games ever made that utilized the microphone, though it isn't clear how advanced the use really was. For example, Super Pitfall II had a special microphone power-up that you could collect and then if you shouted something into the microphone it would activate the power-up. Other games reportedly seemed more sophisticated, but given the computational power and chipset of the Famicom, I would imagine it isn't anything more than detecting that a noise hit the microphone and even then perhaps being able to tell if it is a high/low/hiss/boom type sound.
Finally, the Famicom had a DB15 expansion port connector while the NES had an expansion connector on the bottom of the unit. The Famicom port was more of an extended controller port and thus we saw adapters (I think there was an adapter to allow NES controllers to connect to the unit via this port) and even a keyboard for Famicom BASIC that connected to this. Saving/loading of programs were via cassette audio in/out ports on the keyboard (perhaps related to the microphone's existence?). I believe the keyboard was the only way to get a cassette player hooked up to the Famicom, but despite that extra cost you can see even in the American release of Excitebike! the load/save function for custom built tracks - a feature that required the Famicom's cassette system to operate.
As for the NES connector, I don't believe anything was ever released for it. From the pinouts, it would look like it is a connection to the CPU bus and thus that would validate some rumors of various expansions that could have been available, but none that I've ever known about.
Yes, there was a disk drive and a modem available for the Famicom as well, but those connected either via the cartridge slot (disk) or DB15 port. I did see a disk drive unit in the USA back in the day so can confirm that it did work with the American NES as my friend demonstrated how he could load a NES cartridge from floppy disk.