I have seen the potential refutation of the Danish Gambit, which is not the Schlecter Defense. I have no intention of posting it, but it looks tough to play against. Seems like the King's Gambit is more versatile and gives you better options. That being said, the Danish is just deadly when your opponent doesn't know how to handle it. I find it much easier to crush an unprepared opponent with the Danish.
kings gambit vs danish gambit: which (in your opinion) is the better opening?

come on. if you're gonna make annotations for the games at least make it right.
KG: 1.e4 e5 2.f4(usually Nf3 is followed by nearly any of black's moves, so i'll let you in on that mistake)
DG: 1.e4 e5 2. d4 exd4(this is important) 3. c3(this as well)
there are many different openings after 2.d4, as well as 2. ... exd4(one main one for the latter is the Center game Accepted, which is basically 2. ... exd4 3. Qxd4, which is the main line) and c3 is the danish gambit, although black can refuse to take this and instead do something like d5, and if you decide not to take, try not to take later on either.
To the real topic at hand, King's Gambit is often a lot harder for black to play against in equal elo style.

come on. if you're gonna make annotations for the games at least make it right.
KG: 1.e4 e5 2.f4(usually Nf3 is followed by nearly any of black's moves, so i'll let you in on that mistake)
In general, 3. Bc4 is as good as 3. Nf3.

Danish gambit belongs to professional chess guessing and Kings gambit is street chess and has developed step by step not at one go like the Danish gambit!
Street chess ?!? Oppa Gangnam style!
Hop hop ...

KG is a lot more fun on the accepted siude though, especially to its queen's counterpart.
also, the main point of Nf3 is meant to stop Qh4+, but usually after
3. Bc4 Qh4+ Ke2 white's somewhat down but should be able to hold if s/he studied this line long enough. if not, and black knows how to take the initiative, then you can pretty much say goodbye to winning chances.

The danish gambit 1.e4,e5 2. d4
Which do you believe is better overall? Which is better in rapid?
They are both horrible! Please give me Black against you and promise you will play 2.d4 or 2.f4 and I'd play 1...e5 in a heart beat!
If you want to avoid the theory with 2.Nf3 (i.e. Petroff, Philidor, and something against 2...Nc6, whether that be Scotch, Italian, Ruy Lopez, or the Four Knights Game), then you are better off playing the Vienna (2.Nc3) or possibly the Bishop's Opening. You may not ever get an advantage in the latter case, but at least Black won't already be better by moves 5 to 10!
The danish gambit 1.e4,e5 2. d4
Which do you believe is better overall? Which is better in rapid?