They are homophones. They're homophones in my dialect of AmE. They're homophones according to all the dictionaries I have checked (including the OED), which list /weɪst/ as the IPA pronunciation for both words in both BrE and AmE. And they've been homophones for a long time (see this book from 1818).
There are countless jokes out there that rely on the fact that these words are pronounced the same way:
Except for jokes like these that purposefully blur the line, it is almost always very obvious which one is meant in conversation because of context (and also tone of voice, to some extent). Sometimes the two words are confused in writing, and it's still obvious which is meant because of context.
I am not sure the point of that video you linked that "compares" the pronunciation. (However, although somewhat implied, it does not actually say that there is a difference.)