CHECKSUM computes a 32-bit integer value by quite simple-minded XOR-algorithm. hashbytes uses a more sophisticated algorithm, and the value is longer.
If you want a unique hash value to use for comparison rather than comparing the base values, you cannot use checksum, because the risk for collisions is too big.. This is both due to the simplistic algorithm and the short length of the value. Algorithm aside, if you draw 100000 random 32-bit numbers, the probability for at least one collision is 0.3. And if you draw a million numbers, the probability that all numbers are unique is in the range 1E-50.
I had a case once where I used checksum with some success. There was a table with a number of attributes. There were reason to assume that the same set of value would reappear many times. So I added a checksum column, which I used as one of the keys. If two sets of values would have the same checksum, I had code to handle the collision. But I don't think this is the common case where people ask about checksum.
