How is EdDSA different from ECDSA with a custom curve?
EdDSA is not ECDSA over a different curve. Rather, it is a type of Schnorr signature. Indeed the name is very confusing, and I'm pretty sure that it was chosen in order to give this impression, since Schnorr is less well known.
Schnorr is essentially a zero-knowledge proof of knowledge of the discrete log of the public key, obtained via the Fiat-Shamir paradigm applied to the classic Schnorr Sigma protocol.
ECDSA is a completely different signature scheme, which was designed specifically to bypass Schnorr's patent (at least, this is the understanding in the field).