The big difference between covalent bonds and how rubber-bands behave is in, for lack of a cooler term, stretchiness.
Covalent bonds don't create things that have elastic properties like rubber does. Covalent bonds create things that are very strong, very stiff, and brittle. Diamonds, quartz, and graphite are examples. They're tough, but hit them in the right way and they'll fracture apart. Even diamonds are cut with simple tools, they just have to be cut in the right way.
Rubber is very different. It is able to stretch because it is made up of long, jumbled molecules. Pull on a rubber-band and the jumbled mess straightens out, resulting in stretching. Release it and the molecules go back to their tangled selves.
That's why covalent bonds don't behave like rubber-bands.