Wesley is accepted to Starfleet Academy — this time for real — which means he's leaving the show and this episode should therefore automatically get four stars, right? Kidding, kidding; I don't hate Wesley. At least not always. In seasons three and four he was not nearly as annoying as in previous years. Always too smart, yes, but not as obliviously obnoxious about it. Wesley accompanies Picard for a routine mission, but that mission is interrupted when the broken-down shuttlecraft they're riding in with the pilot (who calls himself "captain") Dirgo (Nick Tate) has a system failure and crashes on a desert moon.
The Enterprise has its hands full on another emergency mission (a disposable procedure plot) and won't reach the crash site for some time, so Picard, Wesley, and Dirgo must in the meantime survive in the desert with no water or supplies.
One of the story's points of labor is that Dirgo is obviously too stupid to live. The first tip-off is that he argues when Picard suggests heading to the mountains, the only possible shelter in sight. The second is that he drinks alcohol in the desert sun. The third is that he fires his phaser into a force field when he very obviously should just wait. That last example comes when the stranded party finds a cave with a fountain of water protected by an automated energy field (a prize behind an obstacle that seems more like the end of a video-game level than something that has a plausibly legitimate reason for being there). This results in a cave-in that critically injures Picard. Wesley must then figure out how to get the water to save Picard's life. Dirgo (as I said, too stupid to live) ends up getting killed behind his own impatient plan that Wesley said was a bad idea — which goes to show that you should never blow off the teenage genius.
The real point of the story is the relationship between Picard and Wesley, and their scenes while Picard appears to be dying. It's heavy on sentiment, gratitude, mutual respect, and the deep-down previously unsaid truth that Wesley considers Picard a surrogate father whom he just hopes will be proud of him. It's earnest, pleasant, intimate — but in the end, "Final Mission" is a little too much like Wesley Crusher: a bit cloying.