The biggest problem probably for Paladin right now is the lack of early game tempo cards compared to other classes. Unless you are playing Murlocs, ~t5 feels awful. This is awful for a class that's based on minion-on-minion attrition combat (lack of a finisher card in the class necessitates this).
Another problem is combo reliant removal and clears. I keep saying this to death, but Paladin has probably the shittiest removal out of all classes, Aldor deals well with minions that rely on stat based combat but fails completely in dealing with minions that rely on effects (e.g. Ysera). Not to mention, if your opponent already has the board, "nullifying" (I say this because it actually doesnt - the minion remains on the board and can ping off your hero power) one of your opponent's minions really doesn't do anything - which comes back to the "I need to keep the board, but I haven't been given tempo minions to do it with" problem. (To be clear what I mean by this is you still need to have minions on the board to kill whatever's been reduced to 1 attack)
High health minions are a complete terror. Dealing with an Ysera or a Malygos when you are even on board is a fucking nightmare. In fact, there's no need to even go that far - empty board on both sides, your opponent slams down Tar Lurker, and the best response you have to that is...?
Equality combos are a good way to clear the board, theoretically, once you have both cards - yet if you have just one part of the piece, they are almost completely dead in your hand. I'm sure if you've played Paladin at all you've come across a situation where you're staring at an enemy board, with double Equality in your hand and no sign of that Consecration/Wild Pyromancer. (Or there's a Murloc Warleader on the board - pre-nerf)
This is partly due to another problem - lack of reliable draw. Hunter shares this problem as well to be honest, as Paladin lacks good draw cards (in standard). To be fair Blizzard have tried to remedy this this expansion by giving us Potion of Heroism which I think is a great cycle card. But what else do we have? Divine Favour is one - although only if you're in an aggro vs control matchup, or at least in matchups where the opponent is slower than yours. Then we have Lay on Hands, which basically might as well read "pass your turn, draw 3 cards." Right now you can't really afford to pass your turn, usually speaking. Otherwise Temporus would be a good card, and the Mage Quest reward would be meh. Again, to be fair this has been partially remedied by Stonehill Defender etc.
I think it's completely weird that they are giving us Healadin synergies when Light Rag and Forbidden Healing, Burnbristle etc are rotating out of standard soon. Hopefully they give us more support soon, otherwise it will be dead almost right off the bat - and anyway, I don't see it being really viable against the late game decks right now (Portal Mage doesn't care how much health you can heal, can OTK you from full either through Apprentice or Alex+Giants, Shadow Priest, Jade Druid etc)
Dude Paladin, well, there aren't enough good efficient dude generators nor dude synergies in standard right now to make it work. Level Up is a complete joke - I've seen people compare this with Bloodlust and try to claim it's okay, but Bloodlust hits all minions and not just recruits, and recruits are fragile 1/1s who get killed off almost as soon as you summon them. In Wild that's a different story as you have Justicar and Muster, but seeing as we're talking about Standard that's irrelevant.
Quest Paladin is a fucking joke. 'nuff said on that one.