I'm a USA lawyer so not hyper familiar (we have no such distinction here) but a barrister is licensed to put on a funny hat or wig and actually speak to/interact with a judge. If you're only a solicitor (you can be both) I believe you're limited to interacting with your client and describing the law for them.

I believe this distinction exists on the premise that barristers, being more independent from their clients, will be more honest and fair when interacting with the judge.