Aircraft design is full of compromises. In case of aircraft nose design, the main factor is to reduce drag. As you yourself noted, the main difference is in flow regime, i.e. subsonic and supersonic. There are different types of drag- skin (or friction) drag, form drag and wave drag (which becomes important only at supersonic speeds.
For subsonic flow, the main drag components are the first two- skin friction and form drag. Here, if we consider the fuselage to be a cylinder with the nose in front (which is a pretty good approximation), the blunt nose configuration gives a lesser drag coefficient (non-dimensional drag) compared to the conical or sharp nosed one. This means that all other things being equal, the blunt nose has lesser drag compared to sharp ones in subsonic flow. In particular, the blunt nose has lesser surface area compared to the sharp, long ones and thus lesser skin friction drag.

Image from Fluid Dynamic Drag by Hoerner
From the above figure, you can see that the blunt nose is actually better compared to a sharp nose as far as drag is concerned.

Image from Fluid Dynamic Drag by Hoerner
However, as the aircraft goes to supersonic flow, the main concern becomes wave drag. Here, the sharp nose keeps the shock wave clear of the aircraft.

Image from wotzup.com
Another thing is that in a sharp section, the cross sectional area varies gradually, which helps in prevention of local acceleration (which can result in local sonic conditions). This is the reason why you see aircraft that fly subsonic over most of their flight regime have blunt noses, while those fly supersonic over most of their flight path have sharp noses.
Of course there are other points to consider too:
The manufacturing cost of the sharp nose is usually greater than a blunt nose (has more area, for one).
A long, sharp nose will affect visibility during landings (Concorde had a droop-down nose because of this). Also, parking in airport ramps is going to be difficult.
A sharp nose will have lesser volume (for mounting radar, for example). As a result, most supersonic aircraft actually have a sightly modified nose section (elliptical or oogival, for example), especially, fighter aircraft.

Image from spaceflightsystems.grc.nasa.gov
As can be seen, an elliptical section with same base diameter and height has double the volume of a conical one. In some cases, where the nose landing gear retracts into the nose (like A350), this will be important.