The modern uses are basically the same but in the christian tradition (and I think inherited from judaism) there is a distinct difference between oaths and vows. Oaths are sworn to others, before god, in order to assure our trustworthiness in respect to some specific endeavor. Vows are promises
to god without condition. So oaths can be sworn based on things other than relationship to god (my honor, my mother's grave) but vows can't in this tradition.
So for example like oath of fealty vs vow of poverty. The difference is inherently religious and kind of subtle and not explicit in most contexts, plus vows aren't very common anymore. But if you read for example a christian marriage vow next to an oath of office you can clearly see the different intent.
And again the meanings are usually the same in general modern use and the religious rite aspect is probably not very important to many people. But when talking about a specific formal oath or vow you can definitely tell that it's two separate rites.