That Spock has faced Half-Breed Discrimination has been hinted at in the original series, but this movie shows he's been faced with it since he was newly born, with even his own father making a disapproving "so human" comment upon being presented with his newborn son.
McCoy's hidden pain — making the decision to take his dying father off life support — is truly moving. Especially the fact that a cure was found soon after, compounding his feelings of guilt and loss. This scene is so heartwrenching (and DeForest Kelleyperforms it so sincerely) that it's considered by most to be the single best thing about the entire movie.
Kirk sitting round the campfire with his friends telling them that he expects to die alone.