Are bonus objectives rewarding?

You will get bonus rewards if you complete the bonus objectives within a quest.

If you finish the mission you a pop-up under your mini-map will first show the normal rewards and afterwards the bonus rewards with a text describing which is which.

You can get extra cash, cosmetic items (heads/skins) and equipment as bonus rewards from quest so it's definitely worth going for those extra rewards.

(Also most bonus objectives can reward you with some interesting dialogue when completing them)

To add to what @Jutschge said above - there are a couple of specific missions where you need to complete the optional objectives if you want to 100% the game.

This is because they unlock hidden areas with Red Chests (and Red Chests are required for 100%)

Off the top of my head, the ones I remember are:

Holy Spirits: To unlock this chest

Powerful Connections: To unlock this chest in Marcus's secret bunker. (This is the quest you mentioned with the spine!)