It's useful to know because in some sectors it can give an indication of which "grade" you might enter a company on. Unless you're in finance it's unlikely if you're applying for a role that a new employer is going to go much beyond a 10% salary increase for the same sort of role. It's then up to you to negotiate your own rate.
What someone's on is a starter for 10. Through the interview process they'll get a feel of whether they think you're worth any extra. Good candidates can always command a premium. It's then down to your or the agency's negotiating skills.
You may want to be paid an amount, they may not want to pay it.
If your commute goes from 15 mins to 2 hours, that's generally not "salaried time". You might want to bump your package up to reflect it, which is where most employers will do current + n% as "can we afford them for the roles we have" (and importantly what grade you enter at - which determines your internal cost and how easy it may be to sell on your services if that's relevant) straight away.
Having said all that, I'd tell the agency what I wanted for any role (assuming your skills are in short supply), and what I'm currently on. There is a credibility / experience / "current perceived worth" view as CBMotorsport says. Believe it or not, it's quite a good filter (in my sector anyway).
As Toys says, current (old employer) is normally checked when you send in your P45.