Let's say you have a friend over for coffee. There are two mugs on the table, not matching - one's a bit bigger than the other. You wouldn't say one mug is "the biggest" because there are only two. "Your cup is the larger one."
Two girls: "The older sister helped the younger one read a book."
Two bowls: "The larger bowl is for the salad. The smaller one is for the soup."
I think it'd be most common to see "the + comparative" when talking about two things.
I'm an American English speaker in the Northeastern US, and I can tell you that in this case, "Your cup is the biggest cup" works just fine and I'd definitely say it sooner than "Your cup is the larger cup", but I'd probably say "Your cup is the bigger cup" less often than "Your cup is the biggest cup", but more often than "Your cup is the larger cup"... Yes, even if there are only two cups.
Confused? To clarify, to me "Your cup is is the biggest cup." sounds the most natural to me, and what would make it even more natural would be changing the second "cup" to "one"... "Your cup is the biggest one" would be what I would usually say, even if there were only two cups.
However, "Your cup is the biggest cup." does not work for this translation, and I keep getting it wrong because I would never say "larger". Now, there is a difference between big, bigger, and biggest in English, so if I were to say "Your cup is the biggest cup." in German, how would I say it?
When you have TWO things which are not the same size then you can say "one [of the two things] is larger than the other. This implies that "one is smaller than the other". When you have three [or more] things then you have things that are "large, larger [second one] and largest [third one]. It is pedantic [correct] to say "The larger of the two things" but some people will say "the largest of two things" which is grammatically incorrect. It is quite common for even English speakers to mix up these two words [larger and largest]. Gilly