I'm possibly biased as I'm from Denmark, but allow me to chip in with my 5 cents around Denmark.
Some of the arguments in here is the 8 vs 16 teams, and that Denmark possibly was the 9th best team:
Yes, possibly - but they'd been on holiday, so their current rank should probably not be that of no 9 in Europe when entering the tournament.
The road to the final, and the level of the opponents:
Denmark's road to the final:
Group:
England (Fielded a bit of a dire team, compared to later teams)
France (A very good team, with Jean Pierre Papin being one of the most highly rated strikers at the time (28 years old in his prime) sided by Cantona, and some other solid players like Blanc, Petit, Deschamps)
Sweden (They went on to become 3rd place in 1994 world cup, so arguably a good team too with Brolin, Andersson and others)
Semi-final: Holland
That is just a supreme team to beat. Van Basten was also FIFA World player of the year in 1992.
Final: Germany
A great team with a Riedle and Klinsmann in front. Riedle went on to score the two decisive goals for Dortmund against Juventus in the Champions League final in 97 and Sammer, Häßler, Effenberg and Kohler was all very good players.
The better team?
Better team? Stars in the team? Possibly Denmark. But there were not many, and it has to be emphasized that team-wise, it was a total different team from the 80s with stars like Elkjær, Morten Olsen and Michael Laudrup. Now the great players were Schmeichel, Brian Laudrup and Flemming Povlsen. This was the team, and not many can honestly say they know more than Brian Laudrup and Peter Schmeichel. Maybe a couple of old-timer United fans remember John Sivebæk, but most of the team was hard-workers. John Jensen was a joke at Arsenal, and was awarded worst player ever signed if I remember correctly. (Scored in the final somehow)
A forgotten player he is Flemming Povlsen, who was a great asset to Dortmund in the 90s before he got injured. Brian Laudrup and Povlsen became the creators in the danish side, but looking at the players beside them they we're basically just really hard workers, probably much like the greek side.
I can't speak for Greece as I don't know them thoroughly, so possibly a greek poster should shed light on that.
The better manager / tactical approach?
Otto Rehagel had a very clear tactical defensive approach, but you can't possibly say the manager in Denmark worked to the team's advantage. Richard Møller Nielsen was simply a weirdo, and started off by having immense talent like Michael Laudrup, Brian Laudrup, Jan Mølby, Elkjær (who dragged Verona to a Scudetto over Juventus) in the 80s.
The danish 80's team had great succes and wowed a lot of people by playing all out attacking football with Elkjær and Laudrup, but suddenly "Ricardo" decided that Michael Laudrup, the godgiven creator of this world, should be chasing balls in a defensively minded side on left wing-back. He was very rigid, and very not-creative, and often set up defensively even against terrible teams, pulling out the creativity of the team.
So he clashed with the Laudrup brothers, which saw Michael (And Brian) leave the team as his talent was simply wasted. I see a comment in here that Michael Laudrup was on holiday due to laziness, but I simply think it was because he had lost all faith in Ricardo's style of football. Rightly so, no one believed in him.
Almost no-one in the squad really wanted to play under Ricardo. As the tournament went on, he sort of just stepped into the background as players played their own way, and the assistent manager helped out more. It was a situation, where Brian Laudrup and Flemming Poulsen decided to be more creative, which they'd not have been if Ricardo had his old way. So as the tournament went on, especially against Netherlands, Denmark became more free-flowing again, allowed players to take chances instead of fitting in for the opponent and going defensively like Ricardo wanted. The total sum of this was some very hard fought games, with nice sparks of creativity.
Summary:
It's hard for me to judge, as I'm biased, and Greece beat some very great teams and more of them. Greece beat France with Zidane, Henry. Portugal with Figo, Ronaldo, and Czech Republic with Nedved (injured), Barosz. Greece had a more defined tactical approach, which was tough for any team to break down, as they worked so well as a unit.
On the other hand Denmark was completely chaotic with a terrible manager and players in bad shape after holidays, so that's a way worse starting point, but they also defeated teams with the best players in their peak: Sweden who became 3rd in 1994 World Cup, A fantastic Netherlands team with Bergkamp and peak Van Basten. And of course Germany with Effenberg, Klinsmann, a country who's just always a winner in finals. It's a bit unfair towards Denmark with a measurement of 8 vs 16 teams imo, as they could just as well have beaten more teams had they had more games. They did however compete against every single of the best teams in that tournament and beat them, so it's a toss up for me. Both were fantastic achievements. Personally (and that's trying to take the being danish part out of the equation) I think Denmark was more enjoyable and entertaining to watch. The Netherland's semi-final was complete gung-ho, and possibly had the entertainment of all Greece's games combined. Huge achievement by such a Greece-squad though. Everyone's recent memory is possibly always the one that is the brightest and best remembered, so they'll possibly pick Greece, but I think it's very close.