These two characters are similar in that they are both unhappy with the society in which they live. They are different, however, because Helmholtz's unhappiness is more principled than Bernard's.
Bernard is unhappy because he can't get into society. Society rejects him because of his looks, so he is not popular. When Bernard gets the chance to enter society's notice because of John the Savage, he takes the chance and acts in the way society expects him to. Bernard hates the idea of being exiled.
Conversely, Helmholtz is unhappy because he dislikes the restrictions society puts on his thoughts and his expression. He is the perfect physical specimen and can easily be accepted by society. Yet he rejects the society because of his principles. When Helmholtz is going to be exiled, he accepts his fate willingly because he truly dislikes the society.
So both men dislike their society, but Helmholtz's objections are principled compared to Bernard's.
Bernard and Helmholtz are similar in that they both feel distressed within their societies. They both express their criticism towards the World State and its operations. They both are unable to fit within their social positions. Their differences come in how they feel the distress. Bernard fights for acceptance by the community while Helmholtz does not really want to belong in that particular community.
The difference between them is that Bernard, even though he fights for acceptance, is weak, unattractive and not as intelligent. Helmholtz is the perfect society’s perfect human specimen; he is strong, attractive, and very intelligent. Bernard is shallow, and he is in pursuit of popularity within his community as seen when he takes full advantage to live the popular life after coming back with John the Savage. Helmholtz is not bothered with popularity or with society’s standards and instead thinks that popularity and those standards are vain.
At the beginning of the novel, Bernard and Helmholtz are already friends, which means they have spent a lot of time together discussing their philosophies on life. Since they are both Alphas, they are very intelligent and need academic stimulation. Bernard works in the Psychology Bureau and Helmholtz works as a writer for Feelies and other unintelligent venues. They are both intelligent enough to have intellectually stimulating conversations and this is why they spend time together. They have recently challenged themselves to act more like adults, rather than children, by restricting their personal impulses. For example, Bernard wants to focus on Lenina, one mate, and Helmholtz wants to restrict all women because they are distracting and he's searching for something profound to write about.
Unfortunately, Bernard's commitment to what he has discussed with his friend is not as strong as Helmholtz's. Once Bernard gets a little bit of popularity, it goes to his head and he becomes the type of person he complained about before. Helmholtz actually pinpoints the similarities and differences between himself and Bernard as follows:
"He liked Bernard; he was grateful to him for being the only man of his acquaintance with whom he could talk about the subjects he felt to be important. Nevertheless, there were things in Bernard which he hated. This boasting, for example. And the outbursts of an abject self-pity with which it alternated. And his deplorable habit of being bold after the event, and full, in absence, of the most extraordinary presence of mind. He hated these things--just because he liked Bernard" (116).
Helmholtz is secure, humble and patient; but Bernard is insecure, arrogant, and pitiful. Not only that, but Helmholtz is tall, handsome, and can have any girl he ever wants, but Bernard is short, eccentric, and not as handsome.