There is obviously a love between them, despite Chlomo's rather distant behavior and Elie's almost succumbing to the dehumanizing effects of the camp and abandoning his father. Both are deeply spiritual, though Chlomo doesn't seem to take Elie's faith as seriously as he might have. They are clearly both aware of their many missed opportunities which Elie always regrets, though his father remains relatively silent about them. In the end, they are both strong and committed to their relationship.
Apart from the differences that clane has identified, remember too that towards the beginning of the novel, Elie tries to persuade his father to liquidate their assets and move to Palestine, like other Jews were doing. Elie's father says in response that he is "too old", perhaps indicating his more traditional outlook on life and his inability to contemplate another life elsewhere. However, as clane indicates, both characters prove themselves to be strong and resilient under the face of unimaginable pressures. However, one of the key factors that perhaps allows them both to survive is that they are together and this unity strengthens them both.
Elie (Eliezer) and his father are similar in that they are both devoutly religious (at least at the beginning of the book). They also both understand their place in the home- they are to study and concern themselves with knowledge and community while the women are to take care of the home. They are both strong spiritually and physically to have survived in the camps as long as they did (even though his father dies, he lived much longer than most men his age). They both look out for one another while they are going through this harrowing experience together.
They are different in that Elie was more sensitive and outward with his emotions than his father. In the beginning of the book Elie describes his father as "unsentimental" and kind of distracted because he noticed very little about his family. While both were strong, Elie proved to be the stronger of the two when it came to having the will to live, bear in mind however that he is much younger than his father as well.