In OU, Lickitung has to wait until the end to do anything, and by then other Pokemon can do a better job. It is in UU that it really shines, and is highly suggested to try. Lickitung is also awesome to look at.
If you look at Lickitung, you can see it has decent HP, but mediocre-to-lousy everything else. It's tied with Porygon for the third lowest total stats for a fully-evolved Normal-type Pokemon. So, what makes it usable? Why should you use it instead of anything else, even in UU? The answer: Swords Dance.
Lickitung is the only pure Normal-type Pokemon that can learn Swords Dance, and it is pretty much a must if you want Lickitung to be useful. It's best to wait for late-game when all the Pokemon are paralyzed so it can set up. Once it gets a couple Swords Dances up, it's all gravy from there. After two Swords Dances, even Exeggutor falls to the mighty Hyper Beam. Body Slam is for when you don't think Hyper Beam can KO yet. Earthquake is for Rock-types and Gengar, obviously.
In UU, it should be your choice if you want a strong sweeper and don't mind setting up a turn. Lickitung's Speed sucks a little more here, since paralysis is used more sparsely, but its stats don't pale as much in comparison to UU Pokemon as they do to OU.
Wrap is always cool for some constriction, and with STAB and Swords Dance, it could actually pose a decent threat. With Screech, you can get to the 4x attack modifier a turn faster (for example, it takes 3 Swords Dances to get to 4x Attack, but with a Swords Dance and Screech, you get your Attack doubled and the opposing Pokemon's Defense halved, still giving your attack 4 times the normal damage). Counter can be nice if you are going against Normal Pokemon in UU, particularly Fearow, and Substitute can help you set up if needed.
Checks and Counters
Exeggutor and Machamp deal with Lickitung pretty easily because they can take hits and deal a lot of damage before Lickitung has set up. Golem and Rhydon, though weak to Earthquake, hardly fear Lickitung's pitiful base 55 Attack. In addition, Slowbro can easily set up on Lickitung.
Taking UU into account, Fighting-types such as Hitmonlee and Primeape make short work of Lickitung. Aerodactyl resists its attacks, although it hates an unlucky paralysis. Again, like Golem and Rhydon, Kabutops and Omastar are a little wary of the super effective Earthquake, but can try and set up alongside Lickitung and take it down.