When you buy two separate tickets, if you miss your connection the airline won't help you. As far as they're concerned, your first flight coming in late is like your car breaking down on the way to the airport - unfortunate, but nothing to do with them. When you buy connecting tickets, once you check in onto the first flight, you're theirs and they will look after you.
Changing planes in Frankfurt, especially from out-of-Schengen to Schengen, takes much longer than at many other airports. (I've actually added it to the list of airports I won't make a connection in.) An hour is a VERY tight connection in Frankfurt - I have done it and had to beg people to let me jump to the front of the line etc, and nearly missed my plane. There is customs, security, and plenty more. A two hour connection would give you enough time in the airport as long as the first flight was not delayed. On separate tickets I would probably only feel secure with a 4 hour connection.
You might have to claim your luggage because of the customs check. Ask your airline.