What is the difference between "nice to see you" and "nice to meet you"?
Are they the same or not?
They are different, "nice to see you" implies that you have already met or seen the person before. "Nice to meet you" means that it is your first time being introduced to that person.
"Nice to see you, it's been a while since we last met."
"Nice to meet you, what's your name?"
Another phrase that is commonly used is "Nice to see you again". It is interchangeable with "Nice to see you".
"Nice to see you again, it's been a while since we last met."
You normally only say "Nice to meet you" the first time you encounter someone. You might reasonably say something like "It was nice to meet [up with] my brother again, after not having seen him for years", but you certainly wouldn't have greeted him with "Nice to meet you [again]".
But "Nice to see you" has no such "first time" connotations. In fact I'd go so far as to say that because meet has such strong associations as given above, discarding it in favour of see actually carries the implicature that you've met the person before.
Others have explained the difference between the two greetings and pointed out how they are normally used.
Nice to meet you
There are at least a couple situations when someone might say this, even though it is not the first time they have encountered or met the person.
1 The speaker is on his way to a meeting where he will meet some new people. Before he gets there, he has an encounter with some unknown person on the street, say the two literary bump into each other, and one spills San Pell all over the shirt of the other. When the speaker arrives at the meeting, guess who is one of the new people he is supposed to meet there? Yes, the guy he encountered earlier. Whether or not the two introduced themselves to each other in their actual first encounter, when they are introduced to each other at the meeting, they could say Nice to meet you, Nice to meet you--again, Nice to meet you, for real, etc. One sees this situation depicted at times in a novel, movie or play.
However, such a situation of 'meeting' someone twice in a short time period does not have to be that dramatic. The two could be introduced to each other more than once within a time period and say Nice to meet you each time. There is the option, of course, to say to the host We've already met.
2 it can also be used when the speaker has not seen or met with the hearer in a very long time (or what feels like a very long time). This usage acknowledges that it has been so long a time since the two people last saw each other that it is similar to meeting for the first time. This would be a humorous but sincere usage. There might be much catching up to do between the two.
Nice meeting you.
Nice seeing you.
In some dialects, at least, one can also use these 'ing' phrases rather than the infinitive phrase. They seem less formal than the infinitive phrases, but are natural to use in a less formal or informal setting. What describes 'formal', 'less formal', 'informal' is local usage. One city or locale may differ from another, and even one business from another in the same city or locale.
In polite society the technical use for a greeting that is first time is "nice to see you." It does sound a bit funny but this is the correct way to respond when you meet someone for first time. It sort of reminds me of the "come again" vs "call again". Though I don't think it would matter that much and doubt most people would pick up on this.