The thing on the left is not usually a called a Büchner funnel, but a frit. Frits are only meant to be used for stuff that can be redissolved afterwards.

You do not put a filter paper or anything on it. The filter grade should be etched onto the glass somewhere. https://en.wikipedia.org/wiki/Fritted_glass

Advantage of the frit is the well defined pore size, you can heat it for drying, it can be mounted in a glass tube with two ground glass connectors (for evacuating), it can be used with concentrated oxidising acids, ...

Advantage of the Büchner funnel is that it does not get clogged with insoluble stuff, ease of use, available in large diameters, very mechanically solid, tolerates alkalic solutions, ...