There are so many differences it's impossible to list them. In general, Postman focuses on having every feature under the sun (integration testing, mock APIs, etc) whereas Insomnia focuses on the core of just being an HTTP client with additional helpers to be more productive. The result being that Insomnia is not suitable for as many use cases but is has a better user experience because it's more focused. Insomnia is also open source software.

You should really try them both to figure out which one best suites you.

P.S. I'm the developer of Insomnia.

I just switched to Insomnia because of a bug in Postman, where it wouldn't pick up changes in the body of an HTTP POST and would use something that had been sent previously. I was able to see that the data received by the server was previous values from the text field. If I cloned the request into a new tab it started working. It happened repeatedly - at some point after making edits to the body it would just get stuck and not send what was in the editor. So I switched because I can't use an app with such issues with its core functionality.

I also disliked Postman's UI, but it wasn't a deal broker. I'm not in love with Insomnia's UI, either, but I like Insomnia because it reliably sends the right request.