ELI5 Aircraft Engines - why propellers on military planes vs jet engines?

Why do military planes like the Hercules C-130 and Airbus A400M military transport planes use propellers instead of jet engines? What makes them more versatile for shorter runways? Why is a turboprop engine not the same as regular piston propeller engines? Explain the difference between a turbojet, turbofan and turboprop.

... we explain like I'm five

Thank you to the r/explainlikeimfive community and in particular the following users whose questions and comments formed the basis of this discussion: elena1995x, dirschau, airbornerodent, frat_guy_pa, arztnur, noxious89123 and kaiisim

To the ELI5 community that has supported us so far, thanks for all your feedback and comments. Join us on Twitter:

https://www.twitter.com/eli5ThePodcast/

or send us an e-mail:

ELI5ThePodcast@gmail.com