Turkish Airlines has a [web page detailing their transfer procedures](http://www.turkishairlines.com/en-int/travel-information/baggage/transit-transfer-passenger-procedures). In your case:

> * International transfer passengers flight routes issued on single ticket continuing to a foreign destination with Turkish Airlines

>   Passengers disembark with only their cabin baggage and proceed directly to the boarding gates without any flight or passport procedures.

>   Passengers who have not obtained their boarding pass for the connecting flight at the initial point of departure must apply to the Turkish Airlines Transit Check-in desk.

> * International transfer passengers (flight routes issued on separate ticket) continuing to a foreign destination with Turkish Airlines*

>   Passengers reclaim their baggage from the international arrivals hall and proceed to the international terminal to apply to the relevant airline counter for check-in procedures.

So if you book separate tickets, you will need to collect your luggage and check it in again. (Seems like an acceptable hassle to me for €120.)

Note that Turkey does not do any passport checks for passengers in transit (staying in the international transit area of the airport), whether they collect their checked luggage or not. You only need a visa if you want to get out of the airport.

In general, booking separate tickets has the downside that if your first flight is delayed and you miss your second flight, the airline disclaims responsibility. With a single ticket, it's always the airline's responsibility to get you to your final destination. In the case of Turkish Airlines, [their rules state](http://www.turkishairlines.com/en-int/travel-information/legal-notice/general-conditions-of-carriage-passenger-and-baggage/schedules-cancellation-of-flight):

> If due to circumstances beyond its control Carrier cancels or delays a flight, is unable to provide previously confirmed space, fails to stop at a passenger's stopover or destination point, or causes the passenger to miss a connecting flight on which the passenger holds a reservation, Carrier shall (…)

I believe that a separate plane ticket (even with the same company) is *not* considered a “connecting flight” here, so with separate tickets you would be on your own. *That* may be worth paying €120 more.