AbdulAlhazred
Legend
You actually have this precisely backwards in the rogue's case. There is literally no balance reason for a rogue not to be able to use a shortbow (a +2/1d8 weapon) when including rogue weapon talent they have access to ranged weapons that are +4/1d4 (dagger) and +3/1d6 (shuriken). The answer is obvious when you look at the rogue's weapon list, however, and is a pure in-universe choice. The rogue's weapons are the following in the PHB: The dagger, the hand crossbow, the shortsword, the shuriken, and the sling. With the arguable exception of the pistol crossbow, every single one of those weapons is concealable. The short bow on the other hand is normally about the length between the bottom of someone's sternum and the ground - unless you're wearing Highlander-style trenchcoats and have Highlander style plot exemption for carrying katanas, there is no way you are going to be able to carry a bow that is longer than your legs around with you without it being spotted and without affecting your walk. The rogue's weapons list has everything to do with in universe justification - what can you carry without being spotted?
And the fighter not wearing plate - I'd argue that +1 armour class for -2 armour check penalty is balanced. So this can't be a balance issue either. It therefore must be intended to reinforce the thematics somehow. And it does this two ways. The first way is that it makes plate armour special without making plate armour ridiculously expensive. In 3.X plate armour was rare and special because it cost 1650GP (did anyone who could afford plate armour not have masterwork plate?) In 4e they wanted to make first level paladins able to be knights in shining armour, so plate armour is officially cheap enough to be bought by a first level PC. On the other hand they wanted to make plate armour rare and special - and couldn't do it through financial cost (a silly way given the D&D economy in any edition). So they did it through opportunity cost - if it (a) costs a feat to wear plate armour even for a fighter and (b) isn't actually better than scale (remember that -2 armour check penalty) so isn't worth the feat then plate armour will be rare and special. And Paladins will get to be our Knights in Shining Armour.
In neither of these cases is "because it's not balanced" even close to being accurate. Both of @Remathilis ' requests are requests for things that are balanced mechanically - but in one case there's a good in-universe reason to not have them (concealment) and in the other it's thematics and worldbuilding.
Of course I'd argue that neither was a good design decision, but for different reasons. The rogue's ability to use a rapier with a feat undermines this theme and both the club and the staff are concealable (or rather the staff can be hidden in plain sight); I'd have had an explicit "Rogue weapon expansion" feat in the PHB if doing all this again, to make this obvious. And the plate armour issue comes under the heading of "Game designers trying to be too clever by half and not explaining why". Something responsible for a lot of problems.
If I were re-writing the rogue weapon proficiencies from the PHB, I'd probably just add the following:
Rogue Weapon Expansion (proposed feat)
Most rogues only carry weapons they can conceal easily, but whether for cultural reasons or through sheer flair you can carry your choice of weapon openly. Gain proficiency with either all simple weapons, or any one one handed martial weapon. You may add this to your rogue's weapon list (and therefore use it with sneak attack and any applicable rogue powers) although a versatile weapon must be wielded in one hand to do sneak attack damage.
Rogue Weapon Master (proposed feat)
Pre-requisite: Rogue Weapon Expansion
Your favoured disguise is that of a warrior, and there's nothing quite like hiding in plain sight. You gain proficiency with all simple and martial weapons and add them to your rogue's weapon list (therefore using them with sneak attack and any applicable rogue powers). When wielding a non-simple weapon in two hands lower your sneak attack damage by 1d6 - this replaces the restriction on versatile weapons for Rogue Weapon Expansion. Further, any weapons the rogue gains proficiency in through superior weapon training feats become rogue weapons.
Shortbow: A rogue may gain proficiency with the shortbow through the Rogue Weapon Expansion feat. Using a shortbow does not lower your sneak attack damage if you also have Rogue Weapon Master.
Whip, Net, Parrying Dagger, Bolas, Talenta Boomerang, Xen'drik Boomerang: The rogue may select the [] as a weapon with the Rogue Weapon Expansion feat
Garrotte: The rogue is automatically proficient with the garrotte.
Kusari-Gama, Spiked Chain: The rogue may select a [] with the Rogue Weapon Expansion feat despite it being a two handed weapon. If they do so, lower sneak attack damage by 1d6.
Why yes, at the price of three feats the rogue can wield an executioner's axe. And this is balanced - the rogue weapon restrictions have everything to do with theme.
4e actually has a LOT of this, WRT weapons. Not only the rogue's weapons, but what about the swordmage for example? There's no reason at all that swordmages shouldn't have access to axes, maces, etc. It isn't even worth a feat. At least the rogue's use of things like rapier and longsword are balanced at the cost of a feat (a battle axe would be equally balanced too though). It was an odd choice too when other classes like the cleric are perfectly welcome to use swords. In the cleric's case there are some nice weapon enchantments for mace that they'll probably want, so it often makes sense for a STR cleric to stick with the more traditional weapon. OTOH even this creates some silliness because it sure seems odd that a cleric of Odin would favor a mace over a spear...
I think the fighter plate armor thing is mostly a matter of attempting to make choices be meaningful even when they're relatively small ones. A fighter CAN benefit from using plate armor. The extra defense is worth a feat, and there are plate options later on, and a few enchantments, that some fighters may well be interested in. It isn't mechanically a wasted feat. However, they could just as easily have not done it and tweaked things a bit differently. Every game design has stuff like this in it though. Design is a process. Choices get made and a lot of the ones made early on have dimensions that aren't appreciated until later, at which point you simply don't have the budget and time to rewrite everything again to try to get rid of them.
PERSONALLY if I were redoing it? I'd greatly simplify weapons in general and make that choice mostly aesthetic. I don't think I like the whole armor proficiency thing either. Let the characters wear what they want to and balance it more cleverly.