Vegeta and Bulma's relationship

Though their relationship initially got off to a rocky start, after several years they became a lot closer, and Vegeta has shown several times that he truly cares about Bulma.

Vegeta's opinion of Bulma
It is revealed that Vegeta became attracted to Bulma because she was strong-willed and overbearing, which most Saiyan males look for. She has been able to argue with Vegeta or even somewhat intimidate him, and this both bothers and attracts him, as most humans, even Chi-Chi, appear to be terrified by his menacing looks and body language.

Vegeta truly cares about Bulma, and does not want her to get hurt. This was revealed when Beerus slapped Bulma in the Battle of Gods Saga in Dragon Ball Super, as Vegeta became enraged and attacked him, who he was fearful of at first. He has grown to fear her temper to some extent, though not to the same extent as Goku is afraid of Chi-Chi. He respects her and will do most of what she says unless he feels he has a clear reason not too. Vegeta also proves to be the only Z-Fighter who can stand up to her directly.

Bulma's opinion of Vegeta
Bulma loves her husband as much as he loves her, and she has shown concern for him whenever he gets hurt. Though Vegeta is somewhat fearful of her temper, on rare occasions the opposite has been true, like when the family was on vacation and Vegeta was sprayed by an octopus, Bulma was worried about Vegeta exploding in anger.

Unlike Chi-Chi with Goku, Bulma does not try to boss around Vegeta. She only has him do things that are reasonable. Even though Vegeta highly respects her, he would not let her boss him around, and this is more of the reason why he hates Chi-Chi because she is even more bossy than Bulma is.