The Ascians are hellbent on destruction because it's the easiest path for them to upset the balance of light and dark enough to rebirth Zodiark as mentioned above.

The Garleans however, are not actually hellbent on destruction. Most of their actual motivation is one of wanting power. Power to destroy the Primals, because they saw what they can do to the world (As mentioned when you end up going to The Burn in the MSQ). With this power, they look to take over the world as a secondary motive, because the further their influence spreads, the more power they gather because of sinister motives such as using civilians as test subjects (For things like the procedures used on Fordola)

Most of what appears to be the Garleans being hungry for destruction, is actually just the Ascians using them to push their own agenda. Be it in the form of promising power to high ranking Garleans (Which again, is what they seek, they want to be able to destroy primals to stop the world from having its aether drained and thus becoming a desolate husk a la The Burn), or by actively masquerading as Garleans, such as taking the body of Zenos.

One of the confusing aspects would be the fact that the Garleans are making deals with the Ascians, while the Ascians are literally aiding beast tribes with summoning the very Primals that the Garleans wish to see destroyed. However, it seems to come down to a similar thing to with the Archbishop in HW where the plan is to go along with them until getting the power to not only deal with the threats they made the alliance for (For the Archbishop, it was to deal with Nidhogg and his horde, for the Garleans it's the Primals) but also then destroy the Ascians themselves.

So, interestingly enough, on reflection, the Garleans and the Warrior of Light and the Scions actually have similar goals in mind. Prevent the Primals from sucking the world dry of aether and to destroy the Ascians. Which is why Alphinaud thought that there was in fact the possibility for a peace treaty with the Garleans. The main division between the Scions and the Garleans, is that the Garleans are prepared to make deals with the Ascians in order to gain the power to deal with the Primals, while the Scions rely on the Echo provided by Hydaelyn. (Well, that and the Garleans totalitarianism regime is not particularly welcome, their discrimination and lack of ethics when it comes to slavery, murder and torture of non-Garlean populations are less than desirable...)