There's the Gothic side, for example. Jane Eyre is considered to be "Gothic", and Eugene Onegin holds elements of it as well. The letter-writing scene has a sense of gloom about it, but more eerie is the dream sequence, with its fantastical creatures and irony. And Byronic heroes? Enter Edward Rochester and Eugene Onegin. Both are former socialites, now living out empty lives in empty ancestral houses. Both despise the high society where they once distinguished themselves. Both believe they've seen all there is to life and human nature.
Tatyana Larin and Jane Eyre are hardly less similar. Tatyana is quiet and plain, keeping her feelings very much to herself. So does Jane. Jane expresses her feelings spontaneously, and so does Tatyana (though more elaborately). They're both of them reclusive, but they're not ashamed of it. And even when they are at odds with society and circumstances, they stay levelheaded and sacrifice their feelings to stay true to their principles.
Why do we still root for Edward & Jane, or Eugene & Tatyana, even when they seem like such polar opposites? I guess everyone will have a different answer to this, but mine is summed up in a word: redemption. Mr Rochester, at the weakest point of his life, repents and finds redemption (though not through Jane, it is important to note). Eugene Onegin, on the other hand, has already gone through a slighter kind of reform before he meets Tatyana. Unfortunately, he either doesn't see it, and/or it's not fully realised, being too obstructed by the lingering pride and old habits left in his life. By the end of the story, Onegin hasn't entirely changed, but his story goes farther than that, and we know he has the potential to change.
It's this redemption that turns Edward into a man worthy to be Jane Eyre's "Mr Right". Likewise, it's the glimmering of this change that makes us hope Onegin will find peace someday.
That is one side of the conflict. The other side is the struggle that the heroine goes through--following her conscience and not her heart. In most other romantic classics, it's the heroine vs. the anti-heroine, or the heroine vs. society, or the heroine vs. her Annoying Relations, etc...in each of these she is a victim. But battling one's own human nature is a much more difficult thing. Jane and Tatyana both pass this ultimate test in their stories; they suffer for it, but they come out the blameless victor in the end.
I think this is the reason I like both of these stories so much. Jane Eyre and Eugene Onegin are true-love romance stories, but just as much, they are about spiritual growth and spiritual strength, and as much about individuals as they are about couples.