Vienna Gambit vs King' Gambit The Vienna game/gambit goes as 1. e4 e5 2. Nc3 .. 3. f4...
Why is this considered much more solid/sound than just the King's gambit. It doesn't seem to me that the extra 2. Nc3 thrown in is doing anything?
Additionally, how is the Vienna gambit different from the Vienna game when 3. g3 is played? 3. g3 followed by fianchetto of the f1 Bishop seems to give more semi-open positions. So it seems to me that 3. f4 and 3. g3 would lead to very very different types of games and should be considered two different openings entirely.
The inclusion of 2. Nc3 ahead of f2-f4 is most significant.
Consider 1. e4 e5 2. f4. Does 2. f4 aid White's development? Does 2. f4 make a threat? In both cases, no. 1. e4 e5 2. f4 isn't a threat because if we gave White an additional move for 3. fxe5, then 3...Qh4+ ruins White's day. That's the first reason for 2. Nc3: by guarding e4, it means f2-f4 will make a genuine threat of f4xe5 because following ...Qd8-h4+, ...Qh4xe4 is out of the picture.
The other reason for 2. Nc3 is its influence on d5, given that when White plays f2-f4 as a means of getting the upper hand in the center (by deflecting the e5-pawn, or capturing it), Black's simplest counter in the center is ...d7-d5. However, it does not at dissuade Black from  2. Nc3 Nf6 3. f4 d5, by far the most common continuation.
If White wants to play this flavor of Vienna, he has to like the positions that result from 4. fxe5 Nxe4 5. Nf3. If White really wants to shut off ...d7-d5, he can postpone f2-f4 for still another move with 3. Bf1-c4. That invites the craziness of 3...Nxe4, which is a whole 'nother ballgame.
2...Nf6 3. f4 d5 is most common, but say Black lets White carry out his plan. Suppose 1. e4 e5 2. Nc3 Nc6 Then 3. f4 exf4 4. Nf3. Black is willing to go for this because the knight on c3 is sort of misplaced!
Again, consider the immediate 2. f4. 1. e4 e5 2. f4 exf4 3. Nf3
(3. Bc4 was the choice of Fischer and Polgar, allowing 3...Qh4+, for whatever that' worth. Keres played 3. Nc3 in his brash youth, after which 3...Qh4+ is more meaningful because White's flight is Ke1-e2 rather than Ke1-f1. Steinitz tried the most thematic 3. d4, but 3...Qh4+ compels 4. Ke2 in that case, also. I wrote an entire book on 3. h4.)
3...g5 4. Bc4
(4. h4 is more positionally logical, though 4. Bc4 is equally popular, partly because some players with Black will choose 4...g4 5. O-O gxf3)
4...Bg7.
4...Bg7 is a huge move, mostly because it guards the h8-rook,. so 5. h4 doesn't force ...g5-g4, but can be met by 5...h6 6. hxg5 hxg5 7. Rxh8 Bxh8.
And since ...Bf8-g7 is such a useful move for Black in those King's Gambit Accepted positions, White would rather leave c3 open for the pawn to support d4 (but that Vienna knight prohibits that, and is therefore somewhat misplaced — how ironic).
About 1. e4 e5 2. Nc3 plus 3. g3. There White is keeping f2-f4 in reserve for a long time, after his kingside development is complete (the Bg2 influences Black's main counterattack square d5 from a distance), and the queenside has d2-d3, after which f2-f4 will have the Bc1 behind it and the  Pg3. These positions are actually very quiet because the pieces don't come into contact as quickly (obviously g2-g3 plus Bf1-g2 is less combative than Bf1-c4 which invites a fight with ...Ng8-f6 plus ...d7-d5). In many of those games, White kept f2-f4 in reserve until it most safe, and then goes on with f4-f5 to take all the kingside space (presumably while Pe4, Nc3, and Bg2 keep Black under wraps).