In a CRC-style code, you're essentially adding up all the bytes and letting it overflow the counter, so that the counter is a fixed size (usually 16 or 32 bits). Then you add a few more bytes, exactly the same size as the counter, so that the data bytes plus the extra bytes add up to zero. The extra bytes are delivered along with the data bytes, so that the recipient can repeat the calculation and verify that the total is still zero. If you modify the data, it is trivial to recalculate the CRC code so that the total is still zero.
Hashes are much, much more complex, and they're non-linear. Each bit of the hash output is intended to depend on every single bit of the input, so that changing a single bit in the input creates a radically different hash output.
In a paper published this year, https://eprint.iacr.org/2020/014.pdf, the authors Gaëtan Leurent and Thomas Peyrin changed the values of 825 bytes out of a 1152 byte PGP key in order to generate a new key with the same signature (aka, the same hash). It only cost about $45k, too.
