What is the difference between symmetric and asymmetric bandwidth?

If a network path is two-way (as most IP connections would be), then the maximum rate of data transfer may be defined in each direction. It is possible... ...that the packets may travel along different routes, going one direction versus the other. Or that devices on the path affect the passage of packets differently in one direction over the other.

Symmetric bandwidth corresponds to the situation when the maximum rate of transfer is the same in both directions. Many (if not most) paths are of this type. This doesn't necessarily mean that the path is symmetric.

Asymmetric bandwidth corresponds to the situation when the maximum rate of transfer is different in each direction. A typical case might be a DSL line with 768 kbps upload and 1.5 Mbps download.