well a turbo back is from the turbo back... a cat back is from the cat back...
turbo back usually consists of down pipe, sports cat or decat, centre sections and back box.
cat back usually consists of centre sections and back box.
cat and down pipe are usually very restrictive so turbo back is better option but obviously costs more... sports cats can be a good few hundred quid for a decent one, decats can be quite cheep but you need a cat to pass your mot so unless you know a friendly mot tester your going to have to put the cat in every year... not sure what the law is if the police had a look at your car and saw you had a decat, unsure if they can do anything? I believe I read somewhere they can fine you? but very unlikely to happen...