What are some ways to increase the romance between two stubborn, but very different characters?
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The best advice I've ever heard about building romantic/sexual tension is to start it from the get-go. The moment your characters meet, there should be a certain je ne sais quoi that unconsciously pulls them towards each other. Even if they are stubborn or in denial about their feelings, people in general have a way of revealing their true emotions without even realizing it. A touch here, a teasing comment there, a moment of absolute forgetfulness when your heroine stops to notice the hero's hair blowing in the wind... Little things like this should snowball throughout the story until neither character can deny their attraction any longer. Just be sure to take it at your characters' pace, and remember that a good love story is one that grows organically between your hero and heroine :)
Romance in books is as diverse as real life, there's no one way about it.
Try imagining your question in terms of a vacation, you can go on a lot of vacations, all vacations are different, you can go to the same vacation but experience it differently, and you can have different ways of experiencing or going to that vacation in the first place.
Character bonding typically occurs when there's a reason for it however, I would highly suggest against just throwing two characters together just for the hell of it - unless you're writing fanfiction or a romance - however even in these cases, romance has an objective that leads the story.
Ask yourself these questions:
Typically by answering these two, you will find that you can come up with a reason to have a romance in the first place, which is something a lot of authors forget - plugging romance 'just because'. If the character is looking to save the world, the reasoning behind having that relationship is that she has some secret key to solving his problem. If the character is struggling with anxiety, then perhaps the romance revolves around trying to break this anxiety.
Once you solve the core root of the romance, you will be able to figure out a path. When writing, I find its better to pick two points and draw a line (that can go everywhere as long as they meet up) rather than draw a single point, and try to find a point to stop at.
They could tease each other, and outside people, and maybe the characters themselves, might think they don't fit well together. Behind the unfriendliness and envy they could feel some devotion. They try to compete with each other. After a while they start to copy behavior. Later, one of the persons can solve a mishap only with the help from the other person. Now the teasing changes to open devotion.