Yes, it is different.
You should use the verb "to see" ("ver": eu vejo, tu vês, ele vê, nós vemos, vós vedes, eles vêem) when you are looking at something like a scene, and you should use the verb "to watch" ("assistir": eu assisto, tu assistes, ele assiste, nós assistimos, vós assistis, eles assistem) when you are watching a TV or a show or a game, etc.
Just to make it more complete: you should use the verb "olhar": eu olho, tu olhas, ele olha, nós olhamos, vós olhais, eles olham, when you are looking at something but not really paying much attention to that.
The English pronoun "you" corresponds to 3 or more different pronouns in other languages. These are single person "you", multiple person "you" and also formal "you" (for singular, plural and so on). For example ver (to see):
- (eu) vejo = I see
- (tu) vês = you see (informal singular "you")
- (você) vê = you see (singular "you", formal in Portugal, informal in Brazil)
- (o senhor/a senhora) vê = you see (formal singular "you", very formal in Portugal)
- (ele/ela) vê = he/she sees
- (nós) vemos = we see
- (vós) vedes = you see (plural "you", rarely used)
- (vocês) veem = you see (informal plural "you")
- (os senhores/as senhoras) veem = you see (formal plural "you")
- (eles/elas) veem
To watch is to pay close attention to something and keep track of its evolution/unfolding. To watch is a time consuming activity.
To see is just to notice with the eyes.
To look at is to point your eyes towards something.
You can watch birds - Assistir aos pássaros (take your time and follow the birds' movements through the sky) - Late edit: observar is better for this bird case. But assistir is better for many cases like "assistir aulas, assistir espetáculos, assistir filmes" and others.
You can see the birds - Ver os pássaros (just notice with your eyes that the birds are there)
You can also look at the birds - Olhar (para) os pássaros (direct your eyes towards the birds)
I think it is hard to translate. First rule is: for movies or theater or class you should use "assistir". Second rule: "ver" and "olhar" are almost the same, but you should use "ver" when you are looking AND paying attention in something; and use "olhar" when your eyes are positioned towards something.
I'm pretty sure it is just as simple as this. Seeing a bird is different than taking the time to watch a bird. This sentence is just referring to seeing a bird, it is different than watching. Portuguese speakers in this post have already said that they would not use the term "watching" to describe the activity of Birdwatching :p