What's the difference between "I see nothing" and "I don't see anything"? In what condition each of them is more appropriate?
There’s no difference in meaning.
Not using an auxiliary verb is
At the Battle of Copenhagen, Nelson may have said [of the signal to retreat, which he could not discern by raising a telescope to his blind eye] “I really do not see the signal.” This has come down to us in popular folklore as “I see no signal”: more poetic, certainly, and possibly more emphatic than even using really.