Nuances: Schatz can mean any of the following: darling, sweetheart, love; an increasingly sickening number of variations on "honey", "doll", "ducky" and "precious" (a la Gollum); a presumably lovely person that you may or may not know but certainly don't harbour the slightest romantic intentions towards; a dear friend; someone you hate and wish to humiliate with fake sugar-sweetness; in Vienna: a person you know absolutely zero about other than that they work with you and you can't quite remember their name (this used to drive me nuts).
Alternatively, it also means "treasure", and oddly enough, though it doesn't work as a direct translation, I find that to be the most fitting of meanings: someone you utterly cherish, and although/because there isn't a word for what you feel that doesn't sound stupid or cheesy, you might as well go all out and stand by how besotted you are.
Roman uses it occasionally, with typical nonchalance, and it strikes me as a very Roman word to use: innocuous and flirtatious and light-hearted if thus intended, but always with that deeper undercurrent of "treasure": meaning, you're mine, and I'll defend my claim to you tooth and claw if I have to.
I find it touching, though probably unintentional on the writers' part, that Deniz uses it first, quite casually, in the
contrived Christmas drama episode; not directly, but referring to Roman when talking to someone else (Nadja). I can't actually see him using it when addressing Roman himself, and I don't remember that ever happening; but I do find it poignant that he would use it to refer to Roman to a third party, without even thinking about it.
When forced to translate it, I'd probably go with "sweetie", although that's a whole other level in English: the fact is just that this is one of the terms that doesn't seem to have a direct equivalent. "Love" is too strong, "darling" too 1956, "honey" too married and homely. Mind you, I'm perfectly fine using any of those for friends, but for a significant other, they just seem too cloyingly sentimental. I'm not entirely fond of "sweetie" either, but at least it seems to have that element of fondly teasing snark that "Schatz" has; not entirely serious, but not entirely flippant either.
...I think too much about these things.
Schatz. I love/hate this word. It combines untapped depths of feeling with the most superficial of off-hand, vaguely fond dismissal, depending on circumstance.