Having written this, I've just noticed Barrie's answer. By far the easiest thing to do is to learn the constructions, but they can be analysed (as Bill Franke asked in a comment). Whether it's actually worth the effort may be a moot point.
Let's deal with the second sentence first, because that's easier.
The second sentence uses the phrasal verb look forward. While this idiomatic verb means anticipate, it behaves grammatically and requires a preposition to indicate the indirect object — what you are looking at, or in this case, looking forward to (it's "to" because it's concerned with time).
Thus look forward requires to as a preposition and a noun or noun phrase. See is a verb and therefore not right; seeing is a gerund behaving as a noun and is therefore the right form to use.
I look forward to seeing you.
The first sentence has a wrinkle or two. Want is not a simple verb!
Where a verb is followed by a second verb, that second verb is always in an infinitive form. Because want is not a modal verb (such as do or can) that infinitive form uses to.
I can have it [can is never followed by a to infinitive]
I want to have it [want is followed by a to infinitive]
Want can also be followed by a noun, or noun phrase, or a gerund which acts as a noun; that is, it can be an ordinary transitive verb.
I want an apple
That last cake wants eating
In this case, seeing you is not a noun phrase or gerund, it's a present participle and thus ungrammatical. Therefore the next word must be an infinitive form of the verb (see) and because of that it needs to.
I want to see you.
The wrinkle is knowing that seeing presented as a choice here is a present participle and not a gerund like the "look forward" sentence. The clue is that it has its own object, you, which conflicts with the use of want. Want fundamentally indicates a need, a lack, which must be satisfied. As such, it can only have a single object, the one thing which satisfies that need, so you can't use seeing you. Thus you must use see and that must be a to infinitive.
Best Answer
We look forward (in thhe sense of anticipating) to a noun or a noun phrase. We do not in native English look forward to a verb. To say the latter is to make a mistake and, no matter how forgivable that may be, it is still a mistake.
Afterthought: it is possible to write prose such as "We look forward to see where we are going" but the sense of looking forward is in this case physical rather than anticipatory.