You can get the same voltage and same power out of each, with the right winding ratios. The advantages I've seen are usually related to how you want the phases referenced to something else.
One advantage of Y is that you get a way to symmetrically reference all three phases to the same voltage (usually earth). If you've got 480VAC line-to-line three-phase AC, that tells you nothing about how far those voltages are from the metal box your electronics are in. If that box is grounded, but the AC lines are all 10 kV away from ground, bad things will happen to your insulation. Tying your neutral to earth allows you to avoid that, and be 100% sure that all three line are within an acceptable voltage of earth at all times.
Having a neutral can also reduce noise, for similar reasons. If the AC lines can suddenly shift relative to the grounded enclosure, that common-mode noise can couple in through parasitic capacitance and wreak havoc on your control and sensing circuits.
And with a neutral you get an obvious defined neutral path for fault, imbalance, or harmonic currents. Those currents having a definite path back to earth means they can be detected more easily, and thus reacted to.
Delta has no obvious grounding location; the AC lines are generally all floating relative to earth. Now, there are exceptions. I've seen corner-grounded systems where one phase is tied to earth. I've seen a center-tap on one phase that's tied to earth. But I think it would be fair to say those are hacks, trying to add a ground reference to what should be a Y transformer, but isn't for historical reasons.
Why would you want to have no reference to earth? Power transmission over long distances. Ground voltage varies from location to location; you can't just tie ground in one building to ground in another building, or you'll have a ground loop and constant current flow through your neutral/ground conductors. If you're only dealing with transmission, and local grounding is expressly not a factor, delta lets you save money by avoiding stringing an extra cable for no good reason.
So the way I usually see things done in an industrial setting is to run power in a delta configuration all the way to the point of use, then transform to Y to get a local earth reference for the equipment.