What is the difference in designing an MPC protocol by ring or field? To the best of my knowledge, most of the MPC protocol's are built on a ring $\mathbb{Z}_{2^\ell}$(usually $\mathbb{Z}_{2^{64}}$) or field $\mathbb{Z}_p$ ($p$ is a big prime), why? What is the difference between them?
Examples:

Ring examples such as ABY, SecureML.

Field example such as SecureGWAS.


Update:
This paper says that SS-MPC over the ring cannot easily construct the constant-round protocol due to a lack of the multiplicative inverse.
[...][Most] of the MPC protocol is built on a ring $\mathbb{Z}_{2^\ell}$(usually $\mathbb{Z}_{2^{64}}$) or field $\mathbb{Z}_p$($p$ is a big prime), why?

The relevant protocols for $\mathbb{Z}_{2^\ell}$ most likely use boolean sharing / boolean garbled circuits (representing each individual bit as a shared value) whereas the ones for $\mathbb{Z}_p$ use arithmetic sharing and in some rare cases arithmetic garbled circuits.
The core difference between these two is the set of operations that can be performed cheaply on each:

Bit-wise XOR tends to be extremely cheap with boolean sharing but is usually prohibitively expensive with arithmetic ones
Normal addition is not too-expensive for boolean sharings but tends to be free for arithmetic ones
Bitwise / -oriented gates and reductions in general tend to be much cheaper for boolean sharings than for arithmetic ones - e.g. equality comparisons or bitwise operations commonly seen in symmetric ciphers
Arithmetic operations tend to be much cheaper in the arithmetic case, especially multiplication which is a primitive operation there but requires $O(\ell^2)$ ish non-free operations for boolean sharings