For that budget I'd probably have to say the Pioneer SP-BS22 speakers (make sure it's 22 and not the 21's), and something like the Topping TP20 amp (though if you can swing it, get the TP22 instead)
Standard price it's a bit more than your budget however, the Pioneers seem to go on sale somewhat frequently for under $100/pr (every once in a great while as low as $65/pr) which puts this suggestion into your budget, just need to be a bit patient with them. I've also seen the Toppings on sale, but it's rare for them to do so from what I've noticed.
As far as being better than the Z623's, that's more subjective and depends on what you want. It'll have better sound quality, but probably won't play as loud (though a stronger amp will take the Pioneers far past what the Logitechs can do). It's hard to tell whether or not the Pioneers will play as low, I only find speculation and completely false numbers for the z623's. My guess is that they will be somewhat close, with the Z623's reaching maybe 5-10hz lower...if even that much.