The Vampire Diaries, a tale of two vampire brothers in love with the same human girl, ran for eight seasons. In the very first episode, Stefan's importance is immediately made apparent, as all the girls are mesmerized by the new kid in school. Only, of course, they don't know that this new kid is a vampire. They also don't know that the recent deaths are due to his wild brother, Damon. Thus, the show starts up this contrast between the brothers: the decent one and the wild one.
This contrast continues with two brothers (Elijah and Klaus) in The Vampire Diaries spin-off, The Originals. Elijah and Stefan parallel each other just like Damon and Klaus. So, there are many similarities between the two characters, although key differences still exist.
Similarity: The Good Brother
Stefan and Elijah are presented as the "good brother." They care about others and have a moral code. When Elijah enters the scene in The Vampire Diaries, Elena automatically trusts him. When Bonnie is against vampires in the earlier seasons, she still sees Stefan as different, better than the rest, someone worthy of trust.
Although their stories become more complicated, they come across as the noble brothers, those who will ultimately do the right thing.
Difference: Age & Maturity
Stefan enters high school masquerading as a teenager. He shows up the football players with his vampire strength. Elijah is beyond this high school display of masculinity. He embraces his age and conducts his life with the dignity of his years.
Elijah wears suits, conducts diplomacy, shows his strength, and also tries to understand others. Stefan is a younger vampire, perhaps still thirsting for what he was denied by becoming a vampire: A normal young adult life.
Similarity: They Never Give Up On Their Brothers
If there is one major flaw that both vampires have (or a strength, depending on someone's perspective) is that they live their lives for their brothers. They see the potential for good in their brothers. Although Stefan hurts when Elena chooses Damon, he reasons that Elena makes Damon a better person. Because of this, he won't let himself mourn the loss of his relationship with Elena.
When we meet Elijah in The Vampire Diaries, he decides to kill Klaus. However, just one morsel of potential goodness from Klaus is enough for Elijah to give Klaus another chance. The number of chances that Elijah has given Klaus over the years is immeasurable, even if Elijah suffers for it. Elijah often puts aside his own happiness for Klaus's.
Difference: Desire for Community
At one point, Elijah created a community of elite vampires, the Strix. He wanted to create a brotherhood that was civil and devoted to creating a better world. It was a lofty goal that, unfortunately, didn't live up to his expectations. However, the very fact that Elijah tries shows he has a larger vision. He wants to make a gift out of his vampire curse and improve the world.
Stefan lives much more in isolation, aside from an occasional friend and his brother. The only reason that he is part of any community in Mystic Falls is due to Elena. On his own, he wouldn't have created solid connections.
Similarity: In Love with the Doppelganger
Stefan was turned into a vampire by Katherine, a doppelganger. He originally is drawn to Elena because she looks like Katherine. Later, we find out that Elena is another doppelganger. So, Stefan is in love with two, although he denies being in love with Katherine (although we can all tell he's lying).
Elijah fell in love with another doppelganger, Tatia. On top of this, Tatia's blood was used by the original witch to turn Elijah and his siblings into vampires, binding the spell. Centuries later, Elijah meets Katherine and falls in love.
So, Stefan and Elijah fell in love twice with a doppelganger. A striking similarity.
Difference: Power
Along with age and maturity, Elijah is more powerful than Stefan. As an original vampire, Elijah is nearly indestructible. In addition, he can compel other vampires, not just humans.
Stefan will never be as powerful as Elijah, even if he were to substitute his alternate blood diet with human blood. While Stefan once outmaneuvered Elijah due to Elijah's ego, Stefan's vampire powers will always be weaker in comparison.
Similarity: Value the Importance of Family
For Elijah, family is always and forever. He would do anything for his family. Matter of fact, Elijah is so tied to his love for his brother and family, that he asked for his memories of them to be taken away when the siblings took in pieces of a dark force (the Hollow) so that the force couldn't become whole. He knew that if he had his memories, he would try to reconnect with his family, defeating the purpose of why they each took a piece of the Hollow.
Stefan values Damon, but he also protects the descendants of his line. In addition, once when he had his humanity turned off, only a visit from his mother worked in getting him to turn it back on.
Difference: Attire
While Stefan does value his "hero hair," as Damon jokingly calls it, he doesn't care as much about his clothing. For Elijah, his suits and generally elegant attire are things that are important to him. In a sense, if he can dress up and present a calm, civil demeanor, then he isn't the beast.
At least, Elijah can pretend that the vampire/beast doesn't live inside him. Besides Stefan's hero hair, he doesn't put as much time or emphasis on the way he dresses. He looks like many of those in Mystic Falls. It could be that Stefan wants to blend in, while Elijah wants to stand out.
Similarity: Willing to Sacrifice Themselves
Stefan and Elijah sacrifice their happiness many times on the shows. They also are willing to sacrifice their lives to save others. In the end, both do exactly that. Elijah sacrifices himself so that Klaus doesn't have to die alone, and he and Klaus die in order to save Hope (Klaus's daughter).
Stefan sacrifices himself to save everyone from being consumed by hellfire. He won't let Damon sacrifice himself because Stefan wants Damon to have a chance at happiness with Elena. Poor Stefan gets married, finds some happiness, develops a better relationship with his brother, and then dies to save everyone.
Difference: Control & the Ripper
Little by little, The Originals reveals that Elijah once had a dark past (one that he hides) where he lost control. However, ever since then, Elijah is the symbol of control. He can hold his hunger and control his power.
Stefan has a much harder time with control. One prick of human blood can turn him into a ripper. When he is a ripper, he literally drains humans and kills them. Stefan can't stop after one sip like Elijah or Damon. In addition, he feeds on the human so intensely that he literally rips their heads off.
When we first meet them both, on the surface it may seem that each has control over their vampire nature. However, the truth is that Elijah mastered control, and Stefan is still (and will always be) struggling to maintain it.