The difference between a sports league logo and a sports team logo is of the scale they each represent. A sports league is a business organization, comprising many sports teams that compete for the winners trophy. To design a sports league logo think in broad terms. The NFL logo and the NBA logo can be your inspiration.
For a sports team logo design, your inspiration needs to be focused on a specific message. Usually, this message is of boldness, strength, and sheer power that sports teams want to convey. Sports logos that contain animal icons such as lions, sharks, and eagles as logos, etc. are famous for the aggressive feel they convey through their designs.