The two have very similar meanings and can both be used to mean "detest, despise, look down upon" but there seems to be a subtle difference.
According to these entries:
看不上 can also carry the meaning of "cannot suit for one’s idea". There's also a blog post that states that 看不起 is more of an objective judgement whereas 看不上 is more subjective, hence the latter could simply mean something is not up to the speaker's standards. However I'm not entirely convinced, and was hoping there were better sources outlining the differences between the two.