Restore External HDD from TM Backup

My External drive with my iPhoto and iTunes libraries died. The drive fell down from a table, and since then it is not recognized -- I thought these kind of drives had shock protection.

Fortunately, I have a TM backup. The question is how to best restore the TM backup. If I give it the same name, will TM recognize it as the same drive, so that it doesn't do a full backup next time, after restore? It is quite a huge amount of data. Judging from some older discusssions, it seems the answer is no, but I would like to check again.