Why do we truncate the hash when using DSA?
During verification you perform $u_1 = H ( m ) w \bmod{q}$ would return the same $u_1$ for  multiple hash values. In general you only want one hash to succeed, even if it is unlikely that an adversary can generate a hash that equals $n q + H(m) w$.

Of course this means that any hash with the leftmost bits identical to the org. hash is also acceptable, but at least you don't have to perform $\bmod{q}$ over an extensively large number.

For modular arithmetic any input (base value for modular exponentiation) larger than the modulus contains spurious information.