BES is a server-side product that accesses the mail database from a local server that receives commands relayed from RIM's NOC. MfE accesses the mail database the same way Windows Mobile does..through Exchange ActiveSync over HTTPS direct. That's primarily client-side, though there is (obviously) a server component And Windows Mobile, while it can theoretically handle multiple folders, isn't always that good at it, especially over slow connections.
Basically, the Blackberry isn't doing anything...it's passing along instructions on what to do to the Blackberry Server program, which is doing the work. Exchange ActiveSync theoretically passes along instructions on what to do as well, but I get the impression that the instruction commands are larger.