What is the difference between a bone density test and a bone mineral density test?

No difference. No difference, unless you are referring to a quantitative bone mineral density CT scan, which is different from a DXA scan. It directly measures density, while DXA calculates it indirectly.
Different objectives. A BMD tests for the density of minerals like Calcium to estimate the strength of your bone and a bone density test determines if you have osteoporosis, a disease which causes the bone to become fragile and more likely to break.