What is the functional difference between stranded and solid-core wire?

Solid wire consists of a single strand or core of wire that is insulated with non-conductive material. Typically, you will find solid core wire in situations where the wire is not designed to be continuously flexed (i.e. your house electrical wiring, etc.)

Stranded wire consists of a bundle of small gauge wires compressed and insulated with non-conductive material. Typically, you will find stranded wires in situations where the wire needs to be routed through tight spaces or experiences frequent flexing or vibration (i.e. pump cable, automotive wire, appliance cables, etc.)

Advantages of solid core wire:

Cheaper to produce
More compact diameter for the same current carrying capability as stranded
Less likely to fail due to corrosion
Easier to terminate (make connections)

Disadvantages of solid core wire:

Typically, only available in small gauges
Continuous flexing or vibration will cause the wire to fatigue and break

Advantages of stranded wire:

Very flexible and withstands a greater amount of flexing and vibration
Easier to route

Disadvantages of stranded wire:

Diameter is larger for the same carrying capability as solid
More costly to produce as the manufacturing process is more complex
More likely to fail due to corrosion from capillary action and a high surface area