Biodegradation is the break down of organic matter by microbes. Bioremediation is an engineered technique applied by people to clean up organic matter by helping microbes with the biodegradation process. One way of doing that is to introduce oxygen into the subsurface to help more aerobic microbes grow in order to clean up oil in the soil.

I imagine that biodegradation and bioremediation overlap a lot, because a lot of remediation involves degrading materials.But some remediation would involve other things, such as removing heavy metals from water. Maybe there are microorganisms that can concentrate heavy metals into themselves, thus removing them from water.

I'm not totally sure about that, but I would guess that biodegradation is the deconstruction of unwanted chemicals by organisms, and that bioremediation is a more general term for environmental remediation by biological agents. However, there may be a technical meaning for these that I do not know.