Are ladybugs and ladybirds the same thing?

Yes, a lightning bug, or as we call it in Tennessee, a fire fly, is a beetle, like a lady bug, which is also a beetle.

They are in different families, but in the same order (beetles, Coleoptera). Lady beetles (or lady bugs, if you wish) are in the Coccinellidae family and lightning bugs are in the Lampyridae family.