The FE is automatic + manual exposure and is battery dependent. The FM is manual exposure only, and the battery runs only the meter.
I bought my FE in 1980, and it's never let me down. I shot it side by side with an F2A for years. Still have both cameras.
I only suggest this, because if you're used to Nikon gear, then it's an easy step. Plus, you can always use your manual focus lenses with the F100 and Nikon digital bodies.
The Olympus cameras are very nice as well. I can also recommend the Pentax MX and the Minolta XD11 with the f/2.0 45mm lens (a nice little package).
Keep in mind that if you buy into any Japanese camera from the 1980s or earlier, you're probably going to have to replace the foam light seals. Plus, the camera might need to be overhauled (cleaned, etc.). You might want to look for a camera that's been serviced.