first of all , nice hand sir!

on the flop you bet he calls ,that basically mean he have any Ax unlikely to be AQ he might four bet you with AQ or raise you on flop, any gutshot ,any hearts draw maybe any Qx, slow play set 33 but raise you on turn may be even some float air ,when he call the turn his range is not change very much but he gonna fold his air ,you check to slow play you full house on the river, the purepose of slow play is to either induce a bluff from air or let him value bet hand worse than you ,in thise case his air isnt' likely to get this river the only air is busted draw ,if he have Ax Qx he probably call but you check he will check down ,if he have busted draw he will bluff some of them,if he have gutshot made straight on turn he will value bet you ,most part of his range gonna be Ax and flush draws,so gutshot is happen to be the case he will value bet you and worse than you , we really looking at is how much he is bluff with his busted draw, given your bet twice ,and three bet before flop, you very likely have a big A to bet twice and may be for some reason to check call the river ,but from his perspect you unlikely to fold a top pair kind of hand ,his bluff probably not that profitable, it's a wired check though, if he unlikely bluff with his busted draw, the only value part is his Ax which you check he likely check back,you bet he will call ,so this check probably not work very well in a vacum,

just my humble oppinion,could be totally wrong .

He said in chat that my check river was awfull.

He s just jealous you had a boat and he didnt :D :P.

I doubt you lose value from AQ here depends on your 3 betting range and how he perceives it but I can imagine he bet it himself a decent %. Also what is relevant is how many chop aces you get here with and how many you think he gets there with. It could very easily be a very nice bluffing spot for you so obv you want this in your range. Betting probably is better because you can get heroed by AT- some and there should nt be THAT many bricked flushes for him.

Probably better to use worse hands or AA for balancing a checking range.

I agree with checking the river not being a good idea. Any hand that will valuebet itself will definitely call the shove as well. That said, you´d only lose "value" to bluffs with missed draws - what should those be? I hardly can imagine any ... by checking you just give him the opportunity to xb with not-too-strrong hands.

I play him because i try to be better than him.

The river check is good if villain can value bet lightly/ turn hands into bluffs or get to the river with a ton of flush draws with no showdown value.

I don't think the check is awful if you have a sample about the villain that leads you to believe he could possible call a 3 bet and two more barrels light and aggressive enough on the river to fire his own. I don't necessarily like his play either though. What else are you telling on the river? Unless you have been getting out of line previously you're really only telling a boat or straight. Making that shove with AQ is not ideal and he has to know that with the way the hand was played that's your range. Unless he isn't as good as you say. Just my two cents. Smart check if you knew he'd bet, but not neg ev on your part.

Repping* auto correct on my phone

I think the chk is fine, in fact, I think its the best play.  It's unlikely that he's calling again w/ KQ and its a tough call for him with Ax.  There's a lot of value in tricking your opponent to vbet AQ or AX here, and getting bluffs from missed FDs.  Unfortunately you have the Jh, so he couldn't have backed into trips here.  If you triple barrel, schmeff probably just calls river with his straight. So I think you played this hand amazing and got max value.  Schmeff is probably just steaming because he got outplayed :)  NH and WP.

With the Jh, and the Jc dead, you block a decent amount of his trips hands, Jxhh, and then KJcc. I doubt he calls turn again with like KJdd, but could be wrong. I think semi readless, I'd just bet here because if he's 4b 6%, he shouldn't have many broadway hands to defend with on this runout, but I'm sure he's knows that, so you can possibly get called by an Axs type hand that would ch back. 

So he can't have many hands that vbet worse than yours, however he does have a lot of KT combos as we block nothing. I'd think checking is only really best if he's going bananas turning Axs/Qx type stuff into bluffs, or you think he's gettin to the river with a lot of FDs that are going to bluff.

Semi readless, I'd say he's going to be calling you a bit light and not doing that bluff stuff enough, so 2-0 in favor of betting yourself.