UmbraLux has the right idea. Different swords EVOLVED differently because they had different purposes when they were developed. For example, the "Crusade" weapons are generally very large swords, very straight, very sharp point, and very heavy. Because when fighting the heavily armored opponent, you are trying to knock them down because it's near impossible to get up in a suite of armor on your own. And then you use the point to finish off the fallen opponent by slipping it between plates, or into the visor. Which is completely different from the katana which was originally developed to decapitate lightly armored soldiers while you were riding horseback... and then later became a symbol of status. Rapier and other light weapons are definitely more of dueling blades and focus on quick attacks and quick deflections.
And while I can see the point of why you don't want to let your PC have a katana, because yes, they are very much overhyped... you're letting your feelings of the weapon (the polar opposite of your PC's) cloud your judgement. I mean, kukri's are just as overhyped in my experience. Let your PC pick whatever weapon (s)he wants. If it is indeed appropriate to mock them, then do so. Also, keep in mind, a decent quality katana (not the cheap "pakastani steel" stuff you normally find) will cost you upwards of $500, while a high quality one will cost your near about $1500, and the very very high quality will run you nearly $10,000 (masumune make). I like katana's personally, and I've done a good amount of research on weapon forging of all kinds. Just remember, 100 folded metal makes for a spectacular explosion of steel (waaaay too brittle to actually use, 10-15 folds is more than plenty).
And yes, mechanically, not much difference. And loss of limb is definitely one of those "extreme consequence" things. That and any "stress" on the physical stress track isn't really damage. It is more of near hits, very very minor cuts, or knicks, or the like. Actual damage is caused only when consequences are taken, or you've taken somebody out.
Oh, and I for one, am much more afraid of the guy with the broadsword than the guy with the katana. Sharp sword means it won't hurt as much, but that broadsword is meant to shatter bones....