What are the differences between Flames, Frost and Shock spells?

Leaving aside the issue of varying resistances that you'll encounter:

As a rule of thumb, Fire spells will do the most damage, including a chance to cause enemies to catch fire and continue to take damage for a few seconds. Frost spells drain opponents stamina, and slow their movement. Shock spells drain Magicka, and can chain to other, nearby opponents.

In addition, there are several perks specific to each damage type. Above and beyond the pure damage increase perks, the Disintegrate, Deep Freeze, and Intense Flames perks add a chance for an additional effect when casting Shock, Frost, or Fire spells on a target, respectively.

Different creatures have different resistances to different types of damage.

Trolls, for instance, take increased damage to fire.

Most undead resist cold.

Many Daedra are vulnerable to shock.

If you focus only on fire spells, for instance, you're going to be out of luck if you wind up fighting a Flame Atronach.

Unlike in Oblivion, where you could basically make any spell you wanted, the spells in Skyrim are immutable, which means additional effects (area vs touch vs stream vs rune) depend on the spell itself. This makes it hard to determine if extra effects are due to the damage type itself (such as setting something on fire -> ongoing fire damage) or just specific to the spell in question.