here is a video of a Katana cutting metal, I know someone said "it couldn't"
Mind ya I do not agree with what they are doing, but it does cut the barrel, I am not saying that it proves the Samurai would win, but just showing a Katana will cut metal. they have destructive test videos on a crap load of swords.
http://www.youtube.com/watch?v=LHm_pJceN5Q
It cut the barrel edge, not the surface. It takes very little to stop a slashing weapon, regardless of how sharp it is. A chain mail is practically impervious to slashes and so is plate. Even hardened leather armor is a serious obstacle.
Samurai themselves did not use the katana as their primary weapon as it is not very effective on a battle field - for that they had the naginata, later replaced by the Yari (spear). I suspect the switch to the Yari was its better ability to pierce armor, while the naginata is basically a slashing blade on a stick. For chopping peasants or dueling another Samurai wearing pajamas the katana is an excellent choise. This is the exact equivalent of medieval knights who carried their arming sword with them at all times, but preferred other weapons on the battlefield as primaries.
By the way, in the movie, the axe the guy is using in comparison with the katana is a regular wood axe. It has little to do with the fighting/battle/war axe - these has a flat and relatively thin blade which could be sharpened like a sword's. Because of the large surface behind the sharp edge, it could sustain much harder strikes then a sword without shattering. Essentially a very wide blade, unlike the broad wedge-shape cross-section of the wood axe that also makes it very heavy and bad for combat.