Sintered vs perforated plate Büchner funnels
The thing on the left is not usually a called a Büchner funnel, but a frit. Frits are generally meant to be used for stuff that can be redissolved afterwards, because they tend to clogg permanently, if very small particles get stuck inside.
You do not normally put a filter paper on it, because that would leak on the edge. The filter grade should be etched onto the glass somewhere. https://en.wikipedia.org/wiki/Fritted_glass Sometimes a frit can be used to hold another, powdery filtering substance, if that has a coarse enough grain size.
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 and/or oxidising acids or other solutions, ...
Advantage of the Büchner funnel is that it does not get clogged with insoluble stuff, ease of use, cheap, available in large diameters, very mechanically solid, tolerates alkalic solutions (glass frits do too, but only briefly), ...