The storm comes as a foretaste of God's wrath against sin - it wakes him up from his foolish rebellion; it causes him to realise his perilous predicament. He refers, in prayer, to his distress (Jonah 2:2), being in the belly of Sheol (Jonah 2:2), being cast into the deep by God [not the sailors] (Jonah 2:3), driven from God's sight (Jonah 2:4), his life fainting away (Jonah 2:7) etc. This prompted him to call to the Lord (Jonah 2:2, Jonah 2:7) and to renew his "vow" (Jonah 2:9).
We've all been to parties or family gatherings where someone's been being teased - and suddenly they snap. They've been pushed just a bit too far and immediately everyone stops teasing them. No-one meant to be malicious and so the teasing stops. What made the difference? I'd suggest that it's the display of anger which lets people know that they've gone beyond the bounds of what's acceptable. The anger causes us to critically review our conduct and often shows that we have been guilty of being unkind and unloving. It produces repentance.
God's anger often works the same way - not least because he is "slow to anger" (as Jonah himself acknowledges in Jonah 4:2). It's never that he's been having a bad day or he's just feeling plain ratty: it's always because our sin merits his anger. Here God's anger is Jonah's wakeup call: we can not run from God and expect there to be no consequences.
And the cross of Jesus should have the same impact on us: if that is what our sin cost; if that is the judgement that it deserves, then how can we continue in it? The punishment that Jesus bore should bring us up short and cause us to consider the seriousness of rebelling against God. God is no fairy - and he is rightly furious by the way we embrace (and encourage by word and deed) rebellion against him; angry with the way we arrogantly assert our own self-determination while depending on him for every breath we breathe.