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What is the difference between "Eu te vi" and "Eu vi você"?

Both phrases mean "I saw you". About "te" and "você": Both are informal, but "Eu te vi" is more informal than "Eu vi você". In the oral speech we use both.

Hmmm !
In my opinion the first one is more correct.
I dont know so many about english but...i think its a kind of object pronoun. Its called "Pronome Oblíquio" in Brazil. And these pronouns are composed by : "pronomes átonos" and "pronomes tônicos". The "pronomes tônicos" need of preposition.
I. E.
* || = preposition and " " = pronoun

Os nossos sonhos estão perdidos |de| "nós" mesmos.
Os homens todos perdem quando dois deles brigam |entre| "si".
Sei o que cabe |a| "mim" fazer.

The more usefull i think is know "pronomes átonos".
I. E.

I told you
eu te disse

Send us an email
Nos envie um email

I kissed her
Eu a beijei

Under have a table to help you.

átono tônico

I ---> me | mim, comigo
You ---> te | ti, contigo
He, she, it ---> se, lhe, o , a | si, consigo, ele, ela
We ---> nos | nós, conosco
You ---> vos | vós, convosco
They ---> se, lhes, os, as | eles, elas, si, consigo

I tryed explain it ! But my english isnt so good ;)
Take care ! :D

Is the same thing, but with diferents words.

É a mema coisa, mas com palavras diferentes.

Actually both mean the same thing...
It´s just that "eu vi você" is a bit more used, as informal...
But if you say 'eu te vi' is totally understandable aswell.

they're interchangeable.
nowadays both are used for formal and informal speech. it usually only depends on what part of brazil you are.
now for the european portuguese, i can't really say much... i just GUESS they use "eu te vi". someone from portugal should answer that though :]

and "eu o/a vi" would be for a third person, wouldn't it? instead of "eu vi ele/ela".

I'd say:

"eu te vi" "eu o vi" "eu a vi" = PORTUGAL

"eu vi você" "eu vi ele" "eu vi ela" = BRASIL

We have the same grammar in Brazil and Portugal, but in real life, brazilians dont speak like the formal grammar.

"eu vi você" "eu vi ele" "eu vi ela" are grammatically incorrect, the correct is "eu te vi" "eu o vi", but we use both to speak.

There is no diffence, same meaning, but if you use "eu te vi" you need to mention the 2nd person as "tu" for example "Eu te vi ontem, onde tu fostes?"

In Speaking Brazilian Portuguese (that includes informal writting language such as e-mail, blog, etc.), the most common is "eu te vi".
The formal grammar don't consider correct because, "te" is the same as "tu" (second person) and in Brazil, we use "voce" (3th person). That means, in writing language, we must write "eu o/a vi". But that sentence relly looks formal. So, sometimes we choose "eu vi voce" instead.