The older first version of the IIc only uses the 5.25 inch drives; the second newer version of the IIc can use both 5.25 inch and 3.5 inch drives.This is because the IWM/SWIM used inside the machine.
To be clear on something: the only 3.5" disk drive the non-Plus IIc can use is the
Unidisk 3.5" Drive, model A2M2053.
(Insert standard reference for telling Apple floppy drives apart) And this has nothing to do with the IWM in the machine.
Wikipedia has a pretty good breakdown on the variations you'll find in the original IIc. The short version is there are basically two major motherboard revisions and three major and several minor ROM revisions. The important distinctions are these:
ROM "255": The original motherboard and ROM 16k combination. This one only supports a single 5.25" floppy drive on the external port, and has no internal expansion capabilities.
ROM 0: Same motherboard as above, but the firmware is now 32k. The major functionality change is the disk port now supports "SmartPort" devices so it can drive the Unidisk 3.5" floppy and various SmartPort hard drives.
ROM 3/4: Firmware feature set is basically the same as above, but the motherboard is a new version that has a connector on it for RAM daughtercards.
If you're buying one you'd ideally like to get a ROM 0 or better, but 255s are the most common. A 255 can be upgraded to ROM 0 but the parts to do so might not be that easy to lay your hands on.
When comparing the IIc to the Plus version the major features you get with the Plus are support for the more common IIgs (and Mac) compatible 3.5" drives and a built-in CPU accelerator so it supports running at 4mhz instead of 1mhz. The CPU accelerator is actually in there in part to support the "dumb" 3.5 drives; the disk controller system used by Apple is very CPU intensive and a 1mhz 65C02 is too slow to handle the data rate of the 3.5" drives directly. (The "Unidisk" drive gets around this problem by having its own CPU and buffer circuitry in it.) Just remember, though, that for most IIe/IIc compatible programs the accelerator is by no means necessary, and in fact will do nothing but muck up most games, so there's a good chance you'd be running the machine in the "slow" mode a lot of the time anyway. Therefore I'd hardly call it a killer feature.
Honestly, if all you care about is running classic Apple II software I'd say that a ROM 0 or 3 IIc is probably the better machine. Most IIe/c-compatible software is on 5.25" disks and it's not always trivial to transfer it to 3.5" (especially when we're talking about self-booting stuff, of which there's a lot), so the built-in 3.5" drive of the Plus might be more of a hassle than a blessing.