Why sweat but not brown?

Onions

The more you cook an onion, the sweeter it is going to get; heat breaks down the volatiles and complex starches and converts them to sugars.

When an onion is completely brown then it is basically caramelized. The point of sweating onions is to draw out some of the pungency, but not all. If you cook them 'til they're brown (caramelized) then they will be very sweet, and not really retain any of that sulfurous "onion-y" flavour at all.

So it's not really a question of how much you're developing the flavours, it's a question of which flavours you're developing. The more sweetness you develop, the more of the original onion flavour you lose.

In my experience, caramelized onions are rarely used as an "ingredient" - they're more of a garnish or side dish, since they wouldn't really impart any significant flavour to the main dish.

Celery and other aromatics

These don't undergo the same drastic flavour changes that onions do, but the principle is the same: You sweat them to draw out moisture and aromas without starting a Maillard or caramelization reaction.

The key is that sweating is a preparation step. Yes, frying/sautéing these for longer would develop more flavour (or at least more of a certain kind of flavour), but you don't want to do that too early, since they're going to sit in the pan for a while longer; sweating means softening them up slightly without eliminating all of the aromas or the natural crunchy texture.

You generally don't brown vegetables during the preparation of a recipe; you only do that if you plan to eat them by themselves without any further cooking. If you brown them, then continue to cook, you will turn them into mush and possibly burn them, and there's no way to fix it after that happens.

It's an interesting question. Without a specific recipe, it's hard to say if either technique is better than the other. You can always caramelize half of the onions and use them with half of the recipe to compare the tastes and see what you prefer.

I say you need to use your judgement to decide if you want that different flavor that caramelizing offers. For instance, are the onions going to be a main or strong secondary flavor or are they just there to create a more complete taste?

Examples:

1. Liver and Onions - The onions are traditionally very caramelized and cooked as the strong, sweet flavor goes nicely with the soft liver.
2. Peppers and onions for hot dogs - The onions (and peppers) are sweat to remove the initial strong flavor. The less cooked flavor pairs better with the hot dog.
3. Stocks/Soups - They just sweat the vegetables to create a subtle vegetable flavor. They're not trying to create a sweet broth.

Also, what about time? You'll probably save yourself a third to half the time to just sweat onions instead of browning them. If all you're aiming to do is remove some of the strong flavors, then just sweat them. Otherwise, try both and see what you like more.