What do Dreams do?

As you've gathered, dreams are chiefly long, atmospheric storylines that provide a small amount of orthogonal content. While you don't need to complete dreams to complete the story, the dreams do give you insight into some of the themes and main storylines of Fallen London.

However:

Completing What The Thunder Said gives you the Stormy-Eyed quality, which does turn up outside dreams.

You can trade progress in Dream stories to Dr Schlomo and, in certain circumstances, the Manager of the Royal Bethlehem.

There are a handful of options in the Cave of the Nadir, that place of dreams and forgetting, that are unlocked by dreams qualities.

Dream progress can be hazardous at high Nightmares, as additional Nightmare-inducing cards will appear in your deck.

Death by Water at a specific level also used to be important for beginning the Seeking Mr Eaten's Name storyline, but no longer. Unfortunately, it is now much easier to set yourself on that path, in the Forgotten Quarter.

For the most part, dreams generate a small amount of stats, items, and qualities along with a big amount of nightmares. I have never found them worthwhile. From what I have been able to tell, they are mostly just a sort of "end game" to grind out. There are spoilers on the wiki and none of them appear to have some vast reward associated with reaching the end of the long dream chains.