Use unique mechanics to reinforce how each mission is different.
If escorting a merchant isfeels the same as escorting a wounded old woman isfeels the same as escorting a prince, then eventually it won't be interesting, no matter how much story you build up around it. Instead, give each situation its own special twist.
Maybe the merchant's horses are skittish, and if the party allows any bandit/monster to get too close to them they'll bolt, dragging the merchant & wagon off into the wilderness, forcing the party to attempt to disengage from their current foes to stay with the merchant. Maybe the old woman can't move very fast, so someone from the party has to give up some of their own actions to help her along and defend her in combat. Maybe the prince is headstrong and gets into trouble easily, so the party periodically has to use social skills to convince him not to wander off and explore random ruins/caves (think of him as an 8-year-old with ADD playing Skyrim).
For McGuffin quests, whenever possible, make it about getting the McGuffin to some specific place rather than going and acquiring the McGuffin (look at Lord of the Rings: it's about Frodo taking the ring to Mt Doom; how did Frodo get the ring? backstory). This allows you to give the McGuffin unique mechanics. Maybe the skull of a necromancer empowers all of its bearer's attacks, but also makes them much more vulnerable to damage; this forces the party to decide between A) giving it to the person who already attacks well so they turn into a real powerhouse and then desperately trying to keep that person alive, or B) giving it to someone who's good at avoiding foes, reducing the risk to the bearer but also reducing the benefit of the skull's power boost. A staff of the winds might shove the bearer 10-15 feet in a random direction at the end of the each of their turns; not an issue on a featureless plain or in a bland square room (places your party should never be), but a definite challenge anywhere that has dangerous edges.
Every mission should complicate the party's preferred tactics in some way. This is easy to do with some kind of penalty, but can also be done with a bonus, as long as the bonus requires certain achievable but not automatic requirements to be met.