With Ulfric holding onto Markarth he gains money, silver, and also keeps all that from the Legion. Ulfric can just ship the silver near Falkreath where there are Stormcloak sympathizers, up the mountains past Helgen, up to Ivarstead and then on to Windhelm. The Silver caravans would be VERY heavily guarded by mercenaries and Stormcloak soldiers. Forsworn may be barberians, but their not stupid. Even if Ulfric did not ship it out of Markarth, he can keep it out of Legion hands, while using the coin to hire mercs in the Reach to attack and harrass Legion forces and caravans leaving Soitude, and by doing that would deny the outlying Legion forces of reinforcements and supplies.
The Legion being in control of the Rift meanwhile would have very little advantages other than being close enough to use it as a staging area for an attack on Windhelm and being able to attack and harrass Stormcloak supply and reinforcement caravans. Riften is a highly corrupted city not unlike Markarth, but Riften has no economical value due to the Thieves guild being based there and bribable guards.
I'm not an Imperial sympathizer and think the war is stupid but I disagree with this.
How can he realistically ship it to Falkreath? The majority of Falkreath is either Neutral or pro Empire. Not to mention that the convoys would be so big that it would be pretty hard to sneak past them without the Legion noticing. All it would take is someone to warn the Legion and the convoy is done. Plus Falkreath+The Rift is a long march back to Windhelm. No matter how many soldiers or mercs Ulfric has to protect the convoys the Forsworn will attack them. The Forsworn wants their land to be theirs again. So they're not cowards and determined. Though it is true he may keep it out of Imperial hands but Ulfric will suffer more from lack of resources then the Empire will. Despite Tullius wanting the Silver.
Now it is true for the Legion controlling Riften but they have more advantages. The Rift Borders Cyrodiil. The Legion could easily send reinforcements from their and establish better communications with Solitude.