What is the main difference between creepers and climbers? Why a creeper can't be a climber?

Both creepers and climbers are weak-stemmed plants and, hence cannot grow erect without support. 

The difference between these two is, creepers spread horizontally along with the soil, whereas climbers take the support of an object for climbing.

Climbers have a very thin, long, and weak stem that cannot stand erect but they can use external support to climb vertically and carry their weight by using their special structures called tendrils.

Examples- Pea, Grapevine, Bitter Gourd, etc.

While

Creepers have very tender, long, thin stems that can neither stand erect nor support all its weight.

Examples- Watermelon, Pumpkin, etc.

Let’s take the example of the creeper, Watermelon to explain why it cannot be a climber.

The sprawling (spreading out over a large area in an untidy or irregular way) growth habit makes watermelon vine’s a creeper, but if you laid up a stick, as support to watermelon it can manage to climb upright but when watermelon fruit becomes heavy then the entire plant falls down.

That's why it creeps on the ground and in creepers category.