Cosine similarity vs The Levenshtein distance
As mentioned in other answers, traditionally cosine is used to measure similarity between vectors whereas Levenshtein is used as a string similarity measure, i.e. measuring the distance between sequences of characters.

Nevertheless they both can be used in non-traditional settings and are indeed comparable:


the vectors compared with cosine can for instance contain frequencies of characters or characters n-grams, hence making it a string similarity measure
one can replace the sequence of characters with a sequence of strings or a sequence of n-grams, thus making Levenshtein a more general distance measure.


The main conceptual difference between Cosine and Levenshtein is that the former assumes a "bag-of-words" vector representation, i.e. compares unordered sets, whereas the latter takes into account the order of the elements in the sequences.

In the context of comparing sequences of words many combinations are possible. In case that's what you're looking for you might be interested in this paper: https://www.aclweb.org/anthology/C08-1075/ (full disclosure: I'm one of the authors).