What is the difference between an ordinary feast and honor feasts?

Honor feasts offer additional gifts to guests attending. You can tell an honor feast on the main feast screen because it will have the golden feast master symbol by the host's name.

Honor feast attempts only become available once you have won the title of 'feast master' by ranking #1 overall in feast scores for the week. The honor feasts are sort of like a way to thank your server for their support in attending your feasts so you could win the title.

The previous answer saying you earn honor feast attempts by hosting 5 standard feasts is not correct.

You have to hold 5 normal feast that week before you can qualify to have an honor feast.