I don't think it's possible to explain the differences between the Hamiltonian and Langrangian approaches without explaining what each one is, and that would be a long answer and would in any case just reproduce the Wikipedia articles you linked to.
If you're serious about learning any form of mathematical physics I'm afraid you need to understand them both. The good news is that they're pretty straightforward. If you can muster enough brain cells to learn quantum field theory you'll have no problems learning Hamiltonian and Langrangian mechanics. Actually, when I first came across them I found them fascinating. I wish I had been taught them at school.
Before you study QFT you should understand very well Classical Field Theory