What are the main differences between these two testing frameworks?
I am a totally new to Test Driven Development and starting from the very beginning.
QUnit is very easy to get started with, as you only need to include two files and a little bit of markup, then you can start writing tests.
Jasmine strength, afaik is its BDD-style syntax, if that is something that you prefer (probably not a selling point for you) and tight integration into Ruby/Rails tools.
In the end both get the job done. I recommend to start with QUnit. Once you're feeling comfortable, try Jasmine and check if the BDD style fits better. If it does and you still want to keep using QUnit, you can add Pavlov to the mix, which provides BDD-style methods for QUnit.