AP Ratios are the same for a given skill throughout all levels, so the main things that Leveling a Skill affect are:

Mana Cost
Base Damage
Cooldown (for some champion skills)
Secondary Effects

Cass' Q is much, much cheaper to spam than her W. Also, while W has higher damage potential than Q if the opponent stays in the cloud, they usually will not, making Q do better damage against Champions, while W is better against creeps, or for zoning (or in late-game committed fights).

Assuming you hit...

Each additional level of Q grants an additional 40 base damage, compared to W's 20 (per 2 seconds), and E's 35.

Q and W both have higher range than E. Q when it hits is guaranteed to last 3 seconds, while E is only guaranteed for 2 seconds (though it can last longer). Since Q has longer range, is more spammable than all except E on an already poisoned target, is much cheaper than her other spells, and has the best guaranteed AP ratio and damage (when it hits) it is generally leveled above the others in the laning phase.

There can be an argument for focusing on E if you're confident you can kill with it, though that is situational.

Both Q and W are better for farming, and Q being cheaper and better for harassing as well, makes it usually the better choice to max out first.

At least that's how I see it.

I enjoy the poison bomb build for Cassiopeia, where you alternate W & Q in the early game, using them to distance harass especially in the 2v2 lane. Makes for a powerful early game and a great opportunity to zone. Miasma is lovely for area denial, and punishing bush huggers.

You completely answered my question. Thank you. Now I understand why Q > W & E.

Thanks for all the help guys. Seriously.