Directional tires are primarily about wet traction, but snow traction is sometimes the intent in the design. A Directional tire tread pattern has a design that forms a "V" or arrow. This design will allow water to push out from the center of the tread and out making this type of tire a good wet traction design.
A Directional tire can easily be identified. All Directional tires will have an arrow on the sidewall of the tire. This arrow tells you how the tire should be mounted on the vehicle.
An Asymmetrical tire has a tread pattern that is designed for two purposes. Asymmetrical tires have one side of the tire that is designed entirely different from the other side of the tire. Asymmetrical tires will always have a marking to identify which side of the tire should be mounted facing out. The sidewall will usually say "mount this side out."
The tire to the right has large blocks that will be on the outside of the tire and smaller blocks on the inside of the tire.
The larger blocks are designed for good dry traction and handling and the smaller blocks are good for wet and snow traction.
Asymmetrical and symmetrical tires can be cross rotated - meaning there are no restrictions to what rotation pattern you can use.