They are brothers born years apart...
The Distressor is a fully featured compressor with just about every comp envelope you could dream of. It does the 1176 and La-xx thing really well (doesn't really quite sound like them, but it can react very similarly), along with subtle compression and extreme blow up the room sounds from it's NUKE settings. The coloration functions on the Distressor are switchable in and out, and they range from subtle to depending on the ratio of the comp, outright aggressive.
The FATSO has a few different types of coloration - "saturation generator" (from what I remember is always ON) which to me sounds a lot like a softly saturated tube, "warmth" function which sounds most (but not completely) like saturated tape, which is able to be dialed in in different amounts, and tranny which is subtle low frequency stuff like you hear when you drive a preamp like a Neve (sort of). There are only four 'best of' compression settings on the FATSO, sometimes they work, sometimes not, IME they work maybe 50% of the time to good effect (hey I'm picky about my compression settings).
The online manuals at Empirical Labs
are very thorough and can explain far more than I can.
They really are quite different, there could be two Distressors and one FATSO in a rack and they won't step on each other's toes more than 20% of the time.