This is one of my favorite game series of all time. I think I played seasons with every single team on the first two versions, some teams more than one season. We had leagues going with the first SNES version and TSB III in college. Man that shit was fun.
More about each game: The original Tecmo Bowl only had four plays per team, no real NFL season, limited team selection. All passes were either complete, or intercepted IIRC.
The next version (TSB on NES) was probably the most played and most popular. Full seasons, stats, etc.
The first SNES version of TSB was very similar, but with better graphics and some minor tweaks. IIRC it also introduced trading. Not sure what else changed, besides rosters.
I've never even seen TSB II Special Edition.
The main difference with TSB III Final Version was that it had created players that you could power up. After numerous seasons, I eventually had a DE that was a god, with virtually all 100s.