Why are there tube sockets with tulip spring and fork spring contacts?

This depends on the intended application. Tulip spring sockets have higher insertion force, but fewer plug-in cycles. They are intended for suspended applications, for example in guitar amplifiers, or for tubes that require a high heater current, such as KT170 or 5U4G. Fork spring contacts, on the other hand, have lower insertion force and allow more plug-in cycles. This makes them well suited for tube testers or for users who like to change their tubes frequently (tube rolling). If you want both properties together, Teflon sockets with gold-plated collet contacts are the best choice.