The simple answer is that the altered version of a note is considered a different note than the unaltered version. In other words, yes, a C and C# are considered two different notes.
To be clear though, the term note actually has three distinct, albeit related meanings in music.
First, the term note can be synonymous with pitch, which appears to be how you are using the word. For instance, if I sing an A and you sing an A then we are singing the same note.
Note can also be used to mean a certain isolated occurance of a pitch, rather than the pitch itself. In this sense, you could say that the first three notes of the theme from the first movement of Beethoven's fifth symphony are the same pitch (G).
Lastly, the term can be used to describe the written notation of a pitch and duration, the actual notes on a page of sheet music.
Assuming you are asking with the first definition in mind, there is actually one more layer of ambiguity. With the examples you mentioned, it is clear that C and C# are two different pitches (they sound at different frequencies). However, most musicians will tell you that C# and Db are two different notes as well, despite the fact that they represent the same pitch (in equal temperment) are are played with the same key on the piano. And then there's also the question of whether or not you are going to say two pitches separated by an octave (two C#s for instance) are the same or different "notes".
The point is that the term note is not a particularly well-defined musical term, which makes your seemingly simple question at lot more complicated than you would think. That said, there is no common definition for which C and C# would ever be considered the same note.