When does a solid sublime and when does it melt?

According to the water phase diagram (linked below), there is region of pressure and temperature combination where water passes directly from solid to vapor. It begins at the triple point (273K, 612 Pa) and continues as the temperature and pressure decrease from that point. Above the triple point, ice will have to pass through the liquid phase before it turns into vapor.

Sublimation and melting are both subject to temperature and pressure.

Ice does melt , but at normal STP does NOT sublime.
Carbon dioxide , the gas does not condense at STP , but directly sublimes to a solid (Dry Ice)

STP ; Standard temperature and pressure ( normal conditions on Earth).

Ice melts at 0(zero) degrees C to water.(liquid)
Carbon dioxide moves from gas to solid (dry ice) directly at -78.5 degrees C and 5.1 atms (510000 Pa)

If the solid has a significant vapour pressure below its melting point, it can be made to sublime by reducing the pressure to the value of the vapour pressure. That sounds complicated but take ice as an example. At -20 deg ice has a vapour pressure of about 100 Pa so by freezing water and applying a pressure of 100Pa or less, by means of a vacuum pump, the ice will cool rapidly to -20 and sublime. This is the basis of the process of freeze drying which is much used in the food industry.

Depends on the material.

Ice melts

Solid CO2 sublimes.