My experience with different weapons is pretty limited, and I dunno how I'd start getting into new weapons this late into the game while learning them properly.
- Lance: Pretty much my main weapon, 3u was my first game and I used to to finish the singleplayer, finish low and high rank port, and almost did some G rank by myself but MH4U came out at that time. I was drawn to it for my preference to polearms, and stuck with it because it can block an insane amount of things which makes it very useful against monsters you have not fought before. The attacks come off pretty quick and if you time it correctly you could do very long combos without having to dodge if you use the counter attack (I was fighting a Kushala Daora with a friend. If I stood underneath him the timing between the wind pressure was just right so that I could attack twice then do a counter against his wind pressure: repeat forever). Only after 4u came out and I saw the gaijin hunter videos did I learn about 'evasion' lancing, which means that people who pick the lance can either be super tanky or super evasive (and quicker than most with evade distance) to a ridiculous degree. On top of all of this there are only a handful of attacks (it's a spear, you poke with it) which means that there are no fancy combos to remember, rather you go by a 3-"beat" rhythm that applies to both attacking and dodging. Oh, and icing on the cake is how in 4u you can mount monsters on flat ground by yourself with the charging jump attack. In multiplayer you pretty much have to avoid everyone as you don't get super armor except for the counter attack. I didn't even -know- super armor existed until I tried other weapons.
- Switchaxe: My experience is fairly limited with this and I've pretty much only used it online. Axe mode has relatively slow but powerful attacks until you start the 'infinite combo' which makes it useful for slower enemies. I dunno, I like the overhead downward strike (That does the most damage right?) but it's fairly slow to pull off without already having done the advancing thrust. Then there's the sword mode which prevents your attacks from bouncing (useful because of SA's pitiful sharpness) and grants an additional effect depending on the phial, typically 20% more damage. The SA I have now has paralysis without Awaken plus an elemental phial which means that not only do I do paralasys with my axe, but in sword mode I do even more paralysis. Top it off with the phial discharge attack and I feel this is a pretty good weapon for applying status effects (Plus getting off that discharge feels good, and the way a weapon feels is pretty important). In the case of paralasys this is very useful in multiplayer as all of your teammates get free reign to whack at the monster for a good 10 seconds or so. I was once in a hunt with randoms where one guy broguht flash bombs and traps. With no further communication we went and fought a (pink?) Rathian. I don't think she got a single attack off the entire fight, spending her entire time flash bombed on the ground, stunned by a hammer, paralyzed by my SA, or stuck in a trap. It was pretty horrifying. So on top of all that you can launch your friends into the air which is fun, but in 4u they can actually attack out of it and mount monsters (or knock them away from danger). The sword mode has a very useful combo that is unlikely to knock your friends around and has a pretty good vertical reach.
- Hunting horn: I've not tried this in singleplayer, but giving buffs to 3 people is extremely useful, especially since you're dealing damage just by selecting which notes to play. It also has a launching attack but I forget which one it is, and it can do stun damage like a hammer. If you've gone through singleplayer depending on buffs from your palicoes then you'll probably want to have at least one horn in your group, because that's basically what a horn does (but more useful and less frustrating). While you move pretty slow with it drawn you can play a 2-note song that makes you move faster(?) than a sword&shield user, very useful for passively dodging and positioning. It can even be useful on harvest tours and the like so you can buff yourself with negate stamina then run around the map collecting what you want.
- Bow: I don't get it. Aren't you guys kinda screwed once you run out of coatings? I mean, a blademaster runs out of sharpness and he spends a few seconds sharpening his weapon. An archer runs out of coatings and... just gets stuck doing normal damage for the rest of the fight? Plus you have to be at just the right distance or you won't even do normal damage. At least you don't bounce, I suppose...?
TLDR: Lance suits two different playstyles depending on the skills you use: super tanky counterattacker or super evasive DPS. Plus a monster with high resist to cut but low resist to blunt will still take as much damage as if he were getting hit with blunt.
Switchaxe has two fun modes that could almost be played independently (though you're missing out if you do). High damage but fairly slow axe mode until you start doing the infinite combo, high damage sword mode that improves power or elemental/status depending on the phial with a phial discharge great for immobile enemies.
Horn: If you liked the buffs Palicoes provide then now you can apply them yourself. All your teammates will love you, even if you don't directly attack monsters (though there's no reason not to, your attacks may be slow but they hit pretty hard).
Bow: Maybe it's because I'm a blademaster... but how can you stand a limited damage potential that could be wasted if you miss or aren't at crit range?
I really apologize for being so long-winded, when I only planned to summarize what I liked about the weapons but...