Higher octane gas is important for some high performance engines, but there is no downside to using higher octane gas other than the cost - and minimal benefits unless your car requires it.
I would also offer that it is becoming more common now that Ethanol (a form of alcohol) is being added to higher octane fuels. There is a downside, from what I am told, about Ethanol. The first is that Ethanol evaporates at a faster rate than "normal" gasoline. You need only worry about this aspect if you're fueling a seasonal engine. (Snowmobile, Lawnmower, boat, etc.)
Physics would suggest also that Ethanol tends to burn hotter. This could be a detriment to air cooled vehicles such as Corvairs and older Volkswagens.
Also, on a car that might be running "cold" or has carbon buildup, a "running on" condition will be SLIGHTLY more likely.