This is what I know, but would be happy to be corrected:

Farcaster requires every relay to store everything that has ever been broadcasted on the protocol. This will have a centralising effect as the network grows and storage requirements increase.

Nostr, on the other hand, has no such requirements. Anyone can run a relay and choose what to store, broadcast and accept and what not to. 

Farcaster takes a different approach to identity and authentication. It depends on Email or Ethereum for it. Nostr has built everything from scratch and has its own system for this. 

The approach followed by Farcaster feels very top-down as there seems to be a team with investors that is driving the direction the protocol takes. 

Nostr on the other hand is quite bottom-up and more open to contributors.