What is the difference in a lumbar compression fracture and a lumbar compression deformity?

They are essentially the same thing. 'Deformity' is often used when it is mild, related to mild chronic trauma, and no fracture lines are seen, because it doesn't sound as bad as a fracture. Compression fracture is often used when it is more severe, when fracture lines are clearly seen, and when is new or related to a sudden or severe trauma.

But they really mean the same thing, the vertebra has been compressed into a smaller height.

Kyphoplasty is a procedure done by some neuroradiologists and interventional radiologists to inject cement into the vertebra to increase or maintain height . This is usually done in people with osteoporosis to prevent worsening fracture. If you are younger and your fracture was from trauma rather than osteoporosis then this wouldn't really be an option.

Often the problem is not the vertebra, but the disc that has been squeezed out by the fracture. An MRI usually shows if this is the problem . If the disc is the problem, and conservative treatment fails, then discectomy by a neurosurgeon is an option. Discograms aren't done too often anymore, but it could be helpful if it isn't clear what is causing your pain. The disco gram might show which disc if any is causing your pain. Often an MRI is adequate to figure this out.

I'm sure a fracture is from stress and a deformity is from birth like birth defect.

Usually they want to fuse them.