I would say the main difference is that vasai is focused on quickly sharing progress in a beautiful view. While loom is a more general-purpose tool made for educators.
Here are other differences I have found.
- loom requires signup/sign in, vasai doesn't
- from first install, loom can be cumbersome to set up your preferences. vasai works out of the box, with 2 clicks to start recording your screen
- loom has a long questionnaire to get the extension (from their website). vasai lets you download the extension straight away
- vasais defaults are better (e.g. no countdown on record)
- loom's playback page is not as elegant as vasai's (vasai removes black borders if the video has any)
- the response time to open the extension is faster compared to loom