An authoring program lets you take some source material, e.g. video, audio, pictures, documents, and create a structure that you want to burn onto some media. This might include adding various audio tracks to your home videos, setting up a menu for a DVD that lets you choose different titles, creating a photo slideshow and so on.
A burning program lets you transfer the finished structure onto recordable/rewritable media e.g. CD-R, DVD+/-R, Blu-ray using an optical drive ("burner").
Some programs are both authoring and burning programs.