Difference between Exalt weapons and regular ones?

EXALT weapons are functionally equivalent to the weapons you have available. After a few months EXALT will upgrade to laser-level weaponry, also identical to the laser weapons you can produce, but without having to use resources to manufacture them. You will need to have researched the equivalent laser weapon to be able to use its EXALT counterpart.

When you keep the EXALT weapons, you might receive an offer from a country to buy some of them. I just received an offer of 30§ for 6 EXALT rocket launchers. Not much, but more than you would get when you sell them immediately.

Tactically, there is no reason to use them (unless you like their looks), as the stats are identical. I can't comment on country requests, as I haven't received one yet.

Note as per @Shadur's comment, that their laser stuff is as good as your laser stuff, and can offer substantial alloy/money savings if you haven't built a full squad's worth by the time EXALT upgrades (though I wouldn't suggest it as a plan).