Partial collisions of hashes, why is this important?
This seems to boil down to whether or not anyone would have a reason to truncate a hash. The answer is yes, absolutely. For example, someone who needs a 256-bit digest may want to use SHA-512/256 instead of SHA-256 on a 64-bit processor for improved performance (the former uses 64-bit operations whereas the latter uses 32-bit operations). SHA-512/256 is nothing more than regular SHA-512 with a distinct IV, truncated to a 256-bit value. Likewise if someone wants a hash with 280 collision resistance, they would want to truncate a stronger hash like SHA-256 to 160 bits instead of using SHA-1 which has cryptographic weaknesses that allow collisions to be found faster than predicted by the birthday bound.

Another reason someone might want to truncate a Merkle–Damgård hash function is to resist length extension attacks. SHA-256 is vulnerable to them, whereas SHA-512/256 is not, despite being of the same size. An efficient collision against such a truncated hash could then be considered problematic.