What are the functional differences between monolithic and disc ceramic capacitors?
If by monolithic, you mean the multi-layer chip caps (sometimes labled MLCC), that's what all the high density ceramic caps are.  

The traditional disc caps are basically just a slab of ceramic with plate on each side, radial leads attached, and dipped in epoxy or maybe ceramic for coating.  These are low capacitance devices (100 or so pF) but can be very high voltage.  Sometimes they are also called safety caps.  

Also, there are disc or cylindrical shaped caps that are really MLCC structures with an electrode around the outside edge and the other electrode in the middle.  These are used as feed through caps in housings of EMI filters, or maybe even as part of a transmission line termination.  

Wikipedia has a pretty good write-up of capacitors here.  

Edit:  

The question of the disk cap and MLCC having the same voltage rating and capacitance and having to choose between the two occupies a very small space in the Venn diagram.  MLCCs were derived from disc technology to overcome limitations in capacitance and allow wider use of ceramic dielectrics which are comparatively easy to fab.  The main things the disk has going for it are the high voltages and rugged construction.  A single wafer of ceramic is going to take more abuse than a stack of thin layers of ceramic.  The choice would come down to things like that.  

If you needed a rugged part, or a part that could not be surface mounted you would choose the disc (I know you can get leaded multilayer ceramics, but its a disappearing market).  If you needed a part with low ESL and more compact size you would choose the MLCC.  It's pretty clear which is usually more desirable.

The uses for disc type caps really are those for which it has an advantage, like high voltages from 3kV to 6kV where you might need a low dissipation coefficient or NPO type stability.  MLCCs don't offer much competition at those higher voltages.