If I write a major scale in the key of A, is there any difference if I write that major scale but change the key to F or G or B or F♯? In other words, are there any differences between different keys or do they feel the same?
A lot of factors go into this answer, but I think instrumentation and tuning system are the most important.
As just one example: if you're writing a piece for a brass quintet, something in B will sound a bit different (often described as "brighter") than something written in B♭. This is due to the mechanics of the instruments and the fact that playing in B requires more tubing throughout the instrument.
As for "feeling": from a listener's standpoint, there's no point in giving a name to that feeling, since everyone's feeling will be different. But it absolutely feels different for the performer. This is especially true, as Tim commented, for vocalists: since they have such a limited range, some transpositions will simply be impossible to sing.
Lastly, even if people don't have absolute pitch, they can spot these differences. When a theme in Wagner is played in D♭, that very same theme in E major sounds different even if you don't have absolute pitch.
If you're writing for piano, you may think that all keys will sound the same, just transposed. But this is only true on an instrument tuned to 12-tone equal temperament; on an instrument with meantone temperament, the transpositions will sound very different, since the distance between each note varies as the key changes.
You ask about an A Major scale; do you mean you want to have the same notes (A B C# D E F# G# A) but change the key signature? Then it will sound exactly the same; you just have to use the proper accidentals.
In any case, any collection of pitches could be found in a piece of any key. A piece "in C" can contain all 12 notes. Even if you only have diatonic notes, C - D - E can be 1 - 2 - 3 in C, 3 - 4 - 5 in A Minor, 4 - 5 - 6 in G, etc, so what key is it in? It's arbitrary.
So there is no difference in the feeling of different keys in isolation, but if there is a modulation, the new key will be heard in relation to the old key. For example, many popular songs move the key up a whole step somewhere in the middle in order to give the song a lift of sorts.
As far as vocalists go, it is the range of the song rather than the key that matters, but of course transposing the key will change the range. If a song's range is g' - g'', the key (pitch collection within that range) shouldn't matter. But once the song is written, changing its key would of course shift the entire range.
Well, I'm inclined to agree with what @Richard and @NeroXIV have answered but I'd like to add also something to these.
At first, as far as tuning is concerned, the sound is very different when one used equal temperament than that of, let's say, just intonation - for more information about these systems and some nice sound clips you can read this and, mainly, this. In some tuning systems some intervals are way too dissonant to be heard in a regular composition, so they are often avoided.
Apart from the voices, almost every instrument produces sound of different quality and colour in different pitches, so altering the initial key of a piece may lead to several differences on its "feeling". Another important aspect that has to do with instruments is the special technique of each instrument. For instance, if one tried to play Biber's 16 Rosary Sonatas all on a violin without scordattura, this would probably lead to either failure or tendinitis, since, many positions in the 14 middle sonatas of this collection are impossible to be played in a regularly tuned violin by an average person. Moreover, tuning a violin in GDAD, for instance, would lead to be able to play an open high D which has a slightly different sound from a D played in any other position.
Also, many traditional folk songs are played in certain keys for technical reasons. For instance, as a greek I know that many folk songs when played with bouzouki or baglamas are always played in D or A so as to take advantage of this instrument's open strings that are used so as to enhance the rhythm, especially in folk dances. So, transposing such a piece in another key that these strings aren't in any of its fundamental chords, then much of the "dancy" feeling of the piece is lost.
Lastly, a personal viewpoint. I tend to consider many scales have not only different "feeling" but also they bring to my mind different colours - and the same applies to several feelings. So, for instance, when listening to B flat I always have something warm and yellow-oragne in my mind even if I don't have a perfect pitch. So, for such people, it might be a little strange listening to a song in B flat, suddenly in A, because, to my mind, for instance, A is more white. And, especially semitone transpositions and not the "usual" male-to-female ones - a whole fifth or so - may sound quite strange to me.
Hope this added something! :)
If I understand the question correctly, you're asking if the key of a piece at least plays a role in determining its mood. There are many websites that list the characteristics of different keys. For example one such says of D Major:
D Major The key of triumph, of Hallejuahs, of war-cries, of victory-rejoicing. Thus, the inviting symphonies, the marches, holiday songs and heaven-rejoicing choruses are set in this key
There is, however, some debate about if and to what extent key impacts on or determines mood. When you talk about mood, in a sense you are describing an emotional response and so plunging yourself into the ridiculously complex world of psychoacoustics.
There are other factors such as instrumentation but I think these have been covered in previous answers.
I firmly believe that each Key has its own musical "color," its own feel. Every time a song is transposed, it always feels different to me. Its energy vibe is either enhanced or subdued, depending upon the new Key. When a song or piece of music is transposed to a new Key, all of its relative pitches and melodic structure are the same, but the frequencies of the pitches being higher or lower with the different Key affects the energy and feel, and the subtle difference between the new Key's scale degrees compared to a different Key also has some effect. But even besides that, each Key is different, and to me listening to the same song in several different Keys has a different vibe and experience; they're not all equal in that respect at all.