It depends on what Black plays. 2...Bc5 or 3...Bc5 against the Vienna or Bishop's Openings causes problems if f4 is played too early, though it can often be played around move 5 to give White a favorable position in a transposed King's Gambit Declined. Black blundered here on move 7, but this miniature gives you a good idea of how explosive the Vienna Game can be:
You can get transpositions between the KGD and the Vienna Gambit. I'm looking at the Vienna Gambit myself at the moment and trying to learn the territory. I believe the only benefit of the Kings Gambit is of surprise value - but then black has many ways of playing. I'd play 1...e5 all the time if I thought white was going to play 2f4 next move.
To make it clear, I believe black should take and hold the pawn - something he can't do with a Vienna move order, except 1e4 e5 2Nc3 Nc6 3f4 and I think black can take here...but otherwise in the Vienna, it seems more rational, lending itself to recurrent themes IMHO, from the white perspective, which the player can familiarise....positions that perhaps occupy a narrower range of general types than in the Kings Gambit...
I am thinking to make Vienna gambit opening my mein opening. Good attacking chances against e5 and 1..c5 GrandPrix
Vienna gambit: Falkbeer variation is the best! Here my win by the checkmate in 12 moves 
:
https://www.chess.com/daily/game/102112357
Probably the Kingsgambit. It leads to equality but with enough changes to outplay weaker players.
1. 2. Lc4 and 3. f4 is just very dubious after the logical black moves 2. .. Nf6 and 3. ... d5. Black is better
2. Vienna with 3. f4 actually runs in the same problem. 2. .. Nf6 and 3. ... d5. And although i would not claim an advantage for black but you get positions where black can get equality in rather simplified positions without too much trouble (or memorization of complex lines)
3. Kingsgambit leads to equal positions too but in many lines becomes very sharp and it will net you a few wins. In the lines that are not sharp there is enough inbalance to outplay weaker opponents (for both white and black !)
Because some people give all kinds of bad moves for black in the Vienna to get you to play it. Normal black players will play more like this.
I switched to the Vienna Game from the Ruy Lopez about a year ago now and it's served me extremely well. The correct response to 1.e4 e5 2.Nc3 Nf6 3.Bc4 Bc5 is not d3 allowing these annoying ...c6 systems (which, incidentally, is why I prefer the Vienna to Bishop's Opening) but rather 4.f4! as recommended in "The Modern Vienna Game" by Ovetchkin and Soloviov (tough reading, but a very thorough analysis of many of the important lines {although it doesn't feature 3.f4 or 3.g3}).
White probably doesn't have a theoretical edge in any of the Vienna Game, Bishop's or the King's Gambit (try finding a strong novelty on around move 16 or further in the Berlin endgame if you're into that stuff), but I believe the Vienna gives the best chances. An example game against the 4...Na5 line (which is generally held to equalise for Black):
Vienna 1.e4, 2.Nc3, 3.f4
Bishops 1.e4, 2. Bc4, 3. f4
Kings 1.e4, 2.f4