The most used toothing system for bevel and hypoid gear are 3: Gleason, Klingelnberg and Oerlikon.
The names come from the machine producer who compete mainly in the market. The last 2 are similar in shape and performances, with Oerlikon standing as a niche product.
Main differences between Gleason and Klingelnberg are to be found in the tooth height and width evolution over the radial path of the tooth itself.
Gear nomenclature
Arguably, the Klingelnberg tooth form is better for high load, or torque applications, this is why it was used in some buses, type 3s: teeth are generally thicker up near the heel of the pinion,
which is where the contact pattern moves to under load.

Gleason Klingelnberg
Working in areas with smaller width (even considering the natural wear after break-in) Working in areas with bigger width
Less expensive (-70/75%) 15-20% more power transmitted
Blades can be done easily in your own facility Blades must be asked to the Klingelnberg in Germany
Easy to correct Hard to correct

In general both exists because, even with price and performance differences, the main users got such a routine that it’s not economically acceptable to change in favor of the other.
For example AgustaWestland prefers Gleason, but FCA passed from Gleason to Klingelnberg on reducers. In F1 the Klingelnberg is more used.
There are many interesting ways to increase the strength if you are at the limit in your calculation:

Original paper