Summer tires can be referred to as "three season tires." They are the best at everything EXCEPT snow, ice and winter temperature, which you'll get in Ohio (for the OP).[/b]
In fact, many "summer" tires perform better in cold temperatures than do many all season tires.
All rubber compounds get harder as the temperature drops, and all get softer as the temperature rises. Up to a point, the softer the tread compound, the more it is like an art gum eraser, the better the car will corner, and the better it will brake. Conversely, when the rubber in the tread compound is very hard, it will skip across the top of the pavement like a well thrown flat stone across the calm surface of a lake in summer.
Most "summer" tires are designed, at least in part, for performance, so they tend to start out with a tread compound that is softer than the tread compound of most all season tires. Most all season tires are designed with a cost-conscious target purchaser, and so they are designed to have good treadwear ratings. The most direct method to make a tire wear more slowly is to make the rubber harder, so all season tires tend to come with harder tread compounds.
There are technologies that allow a tread compound to maintain a fairly uniform hardness over a wide range of operating temperatures. As you might expect, applying those technologies is not cost free, driving up the cost of the tire. Again, those technologies are applied sparingly to all season tires to keep the cost and the price down.
At just above freezing, the tread compounds of most "summer" tires will be softer than the tread compounds of most all season tires. As you go down below that, to below freezing, the treads of the summer tires will get progressively harder, but so will the treads of the all season tires, and the all season tires' tread compounds were harder to begin with. And it is less likely that the all season tires will have the benefit of the temperature range-extending technologies that may have been used in the "summer" tires.
That said, there are some tires that are truly "summer" -- that is, warm weather -- tires. Most racing tires are designed to perform optimally when hot, so they fall in the true "summer" (not three-season) category. Michelin's BFGoodrich subsidiary makes a pair of "summer" tires with the respective suffixes KD and KDW. "KD" stands for "killer dry" and the tire was optimized for the best possible performance on hot dry surfaces. "KDW" stands for "killer dry wet," indicating that some of the dry performance of the KD has been traded off for acceptable wet-road performance. Similarly, at the top of Yokohama's ADVAN line, the ADVAN Neova is really a street-legal racing tire, one that runs at its best when hot, and is very quick under those conditions. But the Neova is a terrible cold weather tire. The Neova's sister tire, the ADVAN Sport, has been engineered to perform well over an exceptionally wide range of temperatures (but it would lose out to the Neova in racing or near-racing situations). I would -- and I
have -- confidently run ADVAN Sports on dry pavement in sub-freezing conditions, expecting (and having my expectations met) better traction under those conditions that I would likely get from any all season tire.