Generally, when you start without experience you should go with the existing solution instead of inventing something new.
However, your situation is different. Neither FATE nor GURPS are frameworks that are designed for browser based RPGs. So the question is about adapting them, not about using them. Adapting an existing tabletop system for an online RPG is non-trivial and likely an overwhelming task for an average person who writes their first online RPG.
If you make your own system, you still have to draw inspiration from somewhere, so I recommend making your own system inspired by FATE or GURPS. That way you can easily make changes that benefit the gameplay and simplify the implementation.