The chance of having a given player act twice in a row punishes people who really overextend their units, or otherwise positions itself poorly, or overcommits to an attack. It makes blitzing things a much more risky proposition, I think. It can also really pay off for the same reasons - if you go twice, you can move your guys around twice and get a better position to charge things, especially with cavalry - but you have to put yourself in huge danger to do it.
I think it adds another layer of consideration to your movements - while 8th edition was a lot like chess in terms of how things moved and charged, and you could set things up turns in advance,
AoS seems to be much more about reading situations as they arise and taking chances that may or may not pay off. Playing cautiously can keep you safe, but taking a risk can make a huge difference - or get one of your units slaughtered.
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p.s. Lovely to see you playing
AoS, sir! I'm hoping to find a game club that plays in Taipei, i'd love to get back into the hobby.