What are the differences between the elliptic curve equations?
All of these are answered by the SafeCurves project:


For twisted Edwards curves, $ax^2 + y^2 \equiv 1 + dx^2y^2 \pmod p$. Edwards curves are the special case a = 1.
Edwards curves can be converted to Montgomery form.Montgomery curves can be converted to Weierstrass form.Some, but not all, Weierstrass curves can be converted to Montgomery form.
The Montgomery ladder (applicable only to Edwards and Montgomery curves) is faster than standard weierstrass point multiplication methods.
the Montgomery ladder is constant-time, while the standard Weierstrass point multiplication methods are not.The Brier-Joye ladder allows constant-time point multiplication on Weierstrass curves, but is much slower than the standard point multiplication methods. The EdDSA signature scheme is only supported by twisted Edwards curves.