At the beginning of Picard's heart surgery, the lead surgeon explains that they will be using a midline approach (which is normal). However, when he begins with the "tissue mitagator," he cuts into Picard's right side.
Trivia: Picard tells Wesley the story of how he got his electronic heart. This story would be revisited in the sixth season episode, "Tapestry."
Wesley asks if something happened before "the Klingons joined the Federation". They never did join.
When Riker decides to take the Enterprise to the location of the distress call, the crew informs him that they will be out of range of the shuttle the captain was on. Why didn't Riker just communicate to the captain about where they were going since they would be in range? This way the captain would know where the Enterprise was going.
Wesley opens a comm channel and says that Shuttle #2 is ready for takeoff. However, in the next scene, when the shuttle is seen powering up, we see "01" on the outside of the shuttle.
Before the conversations in the shuttle we see the stars moving. During the conversations the stars on both sides stand completely still.
In this and several other episodes (that use the same stock footage) the type 7 shuttlecraft that leaves the shuttle bay is slightly different in the interior of the bay than it is in the exterior shot after it departs. Look at the nacelles and the way they connect to the body, as well as having a more boxy, angular shape than the smooth sweeping one.
Watch the sandwich Picard is eating when he gives Wesley a napkin. It varies between a small bite missing, then no bite at all, then a much larger bite, all with Picard never having put it anywhere near his mouth.
Picard: Did you read that book I gave you?
Wesley: Some of it.
Picard: That's reassuring.
Wesley: I just don't have much time.
Picard: There's no greater challenge than the study of philosophy.
Wesley: But, William James won't be on my Starfleet exams.
Picard: The important things never will be. Anyone can be trained in the mechanics of piloting a starship.
Wesley: But, Starfleet Academy...
Picard: It takes more. Open your mind to the past; art, history, philosophy, and all this may mean something.
Picard: (telling Wesley how he'd lost his heart) ...A trio of Nausicaans came in. They were clearly spoiling for a confrontation with a group of fresh-faced Starfleet officers such as ourselves. Well, everyone in the group had the good sense to give these Nausicaans a wide berth, stand off. Everyone, that is except me. I stood toe to toe with the worst of the three, and I told him what I thought of him, his pals, his planet, and I possibly made some passing reference to his questionable parentage...
Footage of the city on Angel One from the episode "Angel One" is recycled here for Starbase 515.
The Pakleds never appear again on Next Generation, although they are mentioned in "Brothers." Allegedly they were too "politically incorrect" in their portrayal of mentally-challenged people. They subsequently appeared in Deep Space Nine in background shots but never with dialogue or anything to distinguish them.
The name "Grebnedlog" is "Goldenberg" backwards.
Picard: Any rumors of my brush with death are greatly exaggerated.
A takeoff of Mark Twain's famous quote in response to his obituary being printed while he was still living: "The rumors of my death are greatly exaggerated."