Most of my recent travel in Europe has been by myself, so an apartment was not really economical, but my last two trips were with someone, so I tried apartments. The first apartment, other than a small kitchen area, was not really any larger than a hotel room. The second apartment was quite a bit larger, with a separate bedroom and bathroom, plus an eat-in kitchen and a living room.
My first observation is that, after provisioning the apartment just for breakfasts - coffee, milk, rolls, cold cuts, cheese, butter, jam, etc, I probably spent as much as I would have for a small hotel room, and more than I would have for a Privatzimmer. Example, I bought a half kilo of coffee and used half. I had to leave the rest behind. I also left half the butter behind.
The apartment was not provided with staples such as sugar or salt - not even a sugar bowel, so I had to buy those and left them behind.
I really like exploring the local eateries, so I didn't eat many evening meals in. I fixed scrambled eggs and bacon one night. Had to buy 10 eggs; left half behind. Another night I fixed schnitzel. Used one egg. Had to buy flour and breading. Left most of those.
So overall, I don't think I saved any money with an apartment. For Germans, who drive and bring supplies from home and take them back, being able to cook in an apartment might be economical, but not for me.