Usually, you don't get a lot of damage, if any, by switching to a lower grade of fuel; the worst you may get is oxygen sensors that burn out a little sooner. But it's rare to hear to serious engine damage, except possibly on ultra-high-performance cars; my personal rule of thumb is if you can afford a $30,000+ car, you car afford to fill it up with right fuel with no financial excuses, since the difference between fuel grades has barely changed at all in the past 15-20 years. Even then, I've never heard of a direct cause of catastrophic failure from using the wrong octane repeatedly.
I had customers that told me they felt the engine response difference in our loaner cars; we used 93 octane, since 91 is rarely available anywhere in Florida. Many customers would use the cheap stuff, or use mid-grade fuel. I'd never promise they'd get more fuel economy (that's setting yourself up for a complaint), but they'd notice a performance difference, and maybe more fuel economy as a bonus.
Performance-wise, I feel a little more lag in medium-to-hard acceleration when using the wrong octane. Some engines pronounce it more than others, the Lexus isn't as sensitive to mid-grade (89) rather than premium (91-93, what the manual says). But my Neon felt much less responsive if I put 87 octane instead of 89 (manual-suggested). Fuel economy suffers more in the Lexus than the Neon with a octane downgrade, interestingly. It's probably all down to the engine computer modifying the fuel-air ratio to make up for the loss.
In short, if your car doesn't ask for anything other than 87 octane, there's no point in upgrading. But if you go to a lesser octane, you're biggest problems are usually just acceleration performance and/or fuel economy.
Note: The fuel here has about 10% ethanol, and my fuel economy has recently suffered a bit from it (which was especially grating when fuel was $4.50/gallon). Lately, I've used the BP "Invigorate", and I've noticed about 20-30 more miles on the tank (I almost always wait until the yellow light turns on). Then again, I've slowed down just a smidge from stoplights. Until I see a 50-mile-per-tank difference, I can't say for sure. It could be the engine has "adjusted" to the added ethanol.