If you are flying on separate tickets, you are not connecting. Those are two entirely separate itineraries.
If you no show for your LH flight, your LH ticket (and any return or onward segments) will be cancelled and retain whatever value they retain under the ticket's fare rules. That is likely $0 and would mean that you would need to purchase a new ticket.
I strongly recommend against flying on two tickets unless you have carefully thought through the consequences. If you do, you should allow for all eventualities because the risks are all yours. Best to fly down the night before or use a carrier which has an interline agreement with LH and will issue a single ticket BUF-JFK-FRA/MUC.