Jesse Tuck is generally okay with being immortal. To a large extent, this is because he likes the idea of being seventeen forever. Despite being the ripe old age of 104, Jesse has the appearance of a handsome, healthy young teenager. It's not hard to see why he's so relaxed about the whole immortality business.
The same cannot be said of Jesse's father, Angus. He positively loathes immortality—so much so that he actually wants to die. It's notable in this regard that when he sees Winnie's grave, he's actually somewhat relieved that she chose not to drink the water that would have made her immortal.
However, Jesse wanted Winnie to drink the enchanted water. This is a further indication that he has no problem with immortality. He has what he regards as a special gift and not unreasonably wants to share that gift with someone he loves. This would explain why he offers Winnie some water from the spring of eternal youth.
That Winnie, as we've seen, chooses not to drink the water, tells us that, when it comes to immortality, she's closer to Angus Tuck than his son, Jesse. She understands that immortality, despite its superficial attractions, is more trouble than it's worth.