Checksums and cryptographic hashes are both concerned with ensuring message integrity - however, they have different strength and are intended to overcome different underlying problems.

Checksums are usually to do with detecting and correcting errors in data transmission. For example, bitcoin addresses use checksums in order to reduce the chance of someone making a typo when entering one (credit card numbers also use checksums for the same reason).

Cryptographic hashes, on the other hand, are about ensuring message integrity against malicious actors - they are used not just in blockchain, but for example in software packages to make sure that you've downloaded the genuine software, and not a copy with a virus or trojan malware inserted.