Hard Mode is unlocked by beating the Boss once. It seems that there are more enemies in Hard Mode, for example the eyes frequently appear in pairs, as well as an alternate version that spawns above the player. Also, enemies that normally don't appear until the second stage of their world will appear in the first stage (i.g. Eyes, Flying Skulls, and Piranhas). Some of the shop item prices are doubled as well. The main difference is the elements that are changed in each world.
In World 1, a new enemy is introduced. Though it is aesthetically different and smaller, they are essentially identical to the swimming Turtles in World 3, since they move back and forth in mid air, can not be shot and must be stomped on.
In World 2, the spike traps function differently. Instead of triggering in a single place when the character lands on it, all the traps activate simultaneously over and over again at a set interval regardless of when or where the player lands. This obviously requires a different strategy for dealing with spike traps.
In World 4, along the World's normal enemies, there are various enemies from the previous worlds, making it more chaotic.
In World 5 the Boss is much harder and shoots two blasts instead of one