Tie Fighter is linear, but missions are more detailed and varied, you won't get strings of search and destroy missions like in WC3 and usually if something weird happens someone (usually your mission officer) will comment over the comms (so no situations like when you first meet asteroid fighters and nobody utters even a mild oath in Wing Commander III).
Bonus and secondary objectives may lead to some special dialogues during the missions and the debriefing for the seconday objectives sometimes offers some extra details regarding the plot.
Basically if you just want to go on with the game only the primary objectives are needed, secondary objectives may give you some extra details on the plot, bonus give extra points and sometimes help with the other objectives. Note that completing any kind of objectives gives points, especially at medium or even hard, it's perfectly possible to achieve the rank of general well before the end of the game on medium. On Easy you get less points which usually means you'll not gain ranks as fast as other difficulty levels.
Sometimes secondary/bonus objectives start little mini-chains of objective, for example: secondary objective only asks to inspect a certain ship then a mission officer asks if you can help disable the ship so it will be captured then enters a transport that will capture the ship and that's a bonus (or "secret") mission objective completed.