In the ASOIAF books there haven't been many battles from the point of view of someone who has fought an army of wights, so it's unclear how they supposed to act or if there has been any evolution of their behavior within the scope of the story.
However we do see plenty of them in Game of Thrones, and in the first few seasons we see the wights act more like actual humans. They walk upright (when they have legs) and when they attack they tend to use their weapons.
One clear occurrence in which we see this is in S04E10 The Children when Bran, Hodor and the Reeds are attacked by undead before they enter the cave. The skeletons fight with weapons, and stand and duel like actual soldiers would.
In S05E08 Hardhome, the Wights act slightly more savagely, attacking with claws and teeth like classic zombies, and occasionally barrelling into people and tackling them with their bodies with little regard for their well-being. They still run around upright and do mostly use weapons to fight, but they already have diverged from being human-like.
Season 6 spoilers:
Then in the episode S06E05 The Door, the wights tend to run around on all fours, climbing on ceilings, and overall being more savage, feral and even less human-like than before. They do still have weapons and are seen in formation when in the presence of the White Walkers, but as soon as they attempt to enter the cave they are seen acting entirely inhumanly.
I suppose the greatest divergence that can be seen is in the way that they execute these two different victims:
In The Children, the skeletons stab Jojen whilst he is lay on the floor defenseless, yet in The Door they not only stab but also maul Leaf (the Forest Child) with claws and teeth when she is in almost the same situation, and they do the same with Hodor (though they don't really have the opportunity to stab him).
Comparing the earlier battle scenes with the most recent ones, they barely seem like the same creatures. Is there a reason, in or out-of universe why they have changed so much?