the one you have bought is a 4" ducted fan. Typically mounted in the loft to blow through a duct. I don't see the air throughput, but they are often in the region of 240 cu.m/hr, which is about three times the nominal output of the common cheap builder's fan. It should make a big improvement. They are usually quiet enough to run for long periods without being annoying, and use very little electricity.
Yours is quite a cheap one, often sold to people with cannabis plantations in the loft, but if it fails, you can get quite good makes next time that will swap in, but cost more.
As your bathroom has no window, the fan can conveniently be wirired to come on with the light switch.
Does your current fan have a run-on timer? If so, you may find an extra wire. Terminate it safely, you may want to use it if you get another Timer fan in future.