Simply because the Empire are acting almost like a shield for Skyrim, defending them from the Aldmeri Dominion. The Stormcloaks are not powerful enough to defend Skyrim were the Thalmor to suddenly come knocking. Yes, the Empire may be the 'puppet' of the Thalmor in Skyrim, but they are also preventing Skyrim from being totally and utterly conquered. They are of course not without their flaws, but politically, they are trying to give Skyrim as good a deal as the circumstances will allow.
Actually, an independent Skyrim has the potential of putting the Thalmor and Dominion in a real pinch, because they can't simply stroll over and beat them into submission. To even invade Skyrim they have 3 options; through Cyrodiil, sail north past Hammerfell and High Rock, or sail to the west and then around and try a landing through Morrowind (or if they feel tremendously stupid, through the icy waters and land somewhere along Skyrim's northern coast). In the case of the Dominion declaring war against Skyrim it would also be fair to assume at the very least Hammerfell would accept if called to arms, even the empire would likely do so in that situation.
So sailing north would be nasty, as they would possibly have to face to combined navies of Hammerfell, High Rock and Skyrim. Even if they manage to prove superior and push through, I doubt the dominion can muster such an enormous force that they can actually get through while having enough ships/troops left to mount any meaningful amphibious assault.
Going through Cyrodiil? Equally nasty. Either they would be fighting the great war all over again with the Empire declaring war along with the seceded regions, and they will have to fight their way all the way up north, and with Hammerfell and Skyrim now independent they can besiege the Imperial City as many times as they'd like, it still won't frighten more than one out of three enemies. The other scenario I can imagine is that the Empire stays neutral and allow the dominion to move armies through their territory unchecked, this however I consider -very- unlikely. Not to mention that if the Empire stays neutral for a few weeks and then declare war the dominion will have their forces surrounded.
Then they can always try to go east/north and go through Morrowind. With friendly states covering the coast along the way that would be their safest bet. I don't know how severe the destruction of mainland Morrowind is though, but if it's a barren wasteland they might also bump into supply issues if there's nothing to scavenge. Not to mention that if another nation of man declares war at this point, they'll be in major trouble with their forces stuck in some ashy podunk at the wrong side of the map.
Not to mention the possible surge of nationalism among Nords, and general revanchism that might get triggered all over the empire (former and current) as a result of "If the nords can tell the dominion, albeit through the empire, to go svck on a nirnroot, then we can too!". And in addition there's always the possibility of trouble at home for the Thalmor. As far as I recall they started out as a political faction, which makes them something akin to the NSDAP in my eyes. And with them fumbling abroad by losing yet another region they previously had control over through a puppet state, and humans proving they're not pushovers, they could run into serious trouble when defending their aggressive super-villain policies.
Point is; an independent Skyrim might force the dominion's hand and push them into something they're not ready for. Effectively turning them into a paper-tiger.