Complexity classes, specifically the ones for algorithms taking asymptotically linear time in the size of their input. One is deterministic, the other nondeterministic.
Other famous classes like L, P or EXP are trivially in their nondeterministic equivalents, but it's unknown if they're strict subsets. In the case of P it's one of the Clay problems. LIN is the main (only?) case where the answer is known, and it's positive.
An example of a problem that's in NP, but we've kind of bet our civilisation on not being in P, is finding a string with a given hash value. It's a function, it gives the same answer every time, but it's preimages are (we think) completely intractable. What I've described as a possible concrete example is a kind of linear-time version of a hash algorithm. That's less useful, but apparently the (linear-time) irreversibility can be proven in that case.