To summarise the Stormcloaks:
Overall Goals:
1: Secession of Skyrim from the Empire
2: Ulfric to become High King of Skyrim
3: Worship of Talos to be re-instated
4: Skyrim to be considered "for the nords" so you can expect no more of this multi racial melting pot stuff.
5: A restoration of more 'traditional' ways of doing things
Empire Goal:
1: Supression of the Illegal uprising
2: Elimination of Ulfric
3: Skyrim to be maintained as part of the Empire
4: Continue to abide by the White-Gold Concordat until they are ready for another war with the Thalmor
Why do the Stormcloaks want to be independent:
Official Answer: Because Nord's should be allowed to rule themselves and worship Talos ( which makes no sense given the ruling arrangements but i won't get into that now)
Unoffcial Answer: Because Ulfric wants to be in charge
Extra Unofficial Answer: Because Ulfric wants to be in charge, a guy with a history of aiding the Thalmor, even after the great war. He's using religion as a way to draw people to its cause.
Why should Syrim stay a part of the Empire:
1: Economic Interdependency-> Skyrim is made rich by trade with the Empire, food and supplies from Cyrodil are utterly neccessary to the economy. Have a look at Sollitude where trade is common for example, and compare it to isolationist Windhelm.
2: Because it's the legal way to go.
3: Because alone, it would be overpowerd by the Thalmor - Seriously, the Empire has a developed military with complex training, logistics and support systems as well as experienced officers. Skyrim is protected by the buffer provided by the legions. If you want an idea of the strength discrepancy, the civil war in Skyrim is one understrength legion against the SC rebels and it's fairly fair unless you intervene with a mild advantage going to the Imperials. Imperial losses during the great war were more than 5 full legions.
4: Because without Skyrim, the Empire will never develop the strength to defeat the Thalmor
5: Because the only legal way for Skyrim to seceed would be for the High King to decide to do so. The high king is elected by the Jarls at the Moot. Ulfric has just decided that he can Murder the high king and then overthrow Jarl's until he's stacked the deck..then he'll call a moot.
6:Because a civil war will devestate Skyrim (and it does, if you side with the SCs, Whiterun (which is neutral until attacked) is overtaken and sacked.
7: Because restoring the worship of Talos temporarily means nothing if the Thalmor over-run you soon afterwards.
8: Because Tiber Septim (talos) formed the Empire to unite mankind. Because Nords essentially founded the Empire and nordic emperor's have ruled it. Because mankind is stronger together than apart and because right now they need to be strong to balance the AD.
9: Because the Empire doesn't 'take everything for itself. During the Great war, cyrodil burned, Imperial blood ran freely and the Imperial city itself was lost. Skyrim benifited from all of that sacrifice (they sent volunteers to join the legion and they paid taxes, sure, but being part of the Empire meant it was Cyrodilic land burning to hold back elves that hated nords as much as they hated any humans.