The thing that Pip and Estella most have in common is their background. Both have humble origins; Pip is an orphan raised by his sister and her blacksmith husband, whereas both of Estella's parents were convicts. The difference, however, is that Estella remains blissfully unaware of her true parentage for many years and therefore feels entitled to treat those from the lower orders like Pip with haughtiness and contempt.

When Pip comes into some money through the good offices of Abel Magwitch, he too is able to escape his humble origins. But like Estella, he cannot escape them for long. When Abel Magwitch is apprehended by the authorities, Pip is no longer able to live the life of a young gentleman about town and so must return, with his tail between his legs, to live with Joe Gargery and his new wife in their humble cottage on the Romney Marshes.

As for poor old Estella, when her scandalous origins are revealed, her dreams of being a lady of quality are immediately dashed. With both Miss Havisham and her brute of a husband no longer alive, she has no one else to turn to except Pip. Despite her cold personality and the cruel, manipulative way she's played Pip, it turns out that Estella, like Pip, is achingly vulnerable and needs to be loved as much as anyone. It is this mutual need, more than anything else, that finally draws them together.

Pip and Estella are alike in that they both have their characters influenced by Miss Havisham. In Estella's case, Miss Havisham has deliberately raised Estella to be cold and proud so that she will break men's hearts. Miss Havisham does this because she wants revenge on all men after being deserted at the bridal altar.

In Pip's case, Miss Havisham inadvertently influences Pip's character. She invites him over to her house as a child so that he will get to know Estella. Her goal is for Pip to fall in love with Estella so that his heart will be broken. This happens, but what also happens is that Pip finds out he has a secret benefactor who is financing his becoming a gentleman. He jumps to the conclusion that it is Miss Havisham, thinking the older woman wants him to marry Estella. Pip, like Estella, becomes very proud and snobbish once his expectations are raised, and he becomes ashamed of the humble people, like Joe, who have been good to him.

Both Estella and Pip end up humbled, Estella through an abusive marriage and Pip when he discovers, to his horror, that his benefactor is the convict Magwitch.

Although both Pip and Estella suffer from snobbish pride that is humbled, they also have different personalities. Estella has been brought up from the start of life to be passive, cold, and aloof. Before he comes into his great expectations, however, Pip is an active, warmhearted, and generous boy.

To compare and contrast Pip and Estella in Charles Dickens' Great Expectations, one must look at how Dickens portrays their external circumstances as well as how he portrays their emotions.
Estella is portrayed as embodying external beauty and how aspirations towards the manners and morals of the English upper classes can warp character. Miss Havisham, in response to her own mistreatment by men has brought Estella up to distrust and both use and abuse men.Estella is aware of her own emotional emptiness; she says of herself:

I have not bestowed my tenderness anywhere. I have never had any such thing." (2.29.75)

Pip is the protagonist of this novel, and the most sympathetic character. Although he is by nature empathetic and kind, his association with Estella and Miss Havisham temporarily warps his character, leading him to sacrifice his better nature on the altar of the British class system, but as he matures towards the end of the novel, and discovers his true parentage, he transcends the narrow and status conscious world of Miss Havisham.

This is a really interesting question to consider. You might want to think about Chapter Eight, which is our first introduction to Estella, as a good passage to compare both of these characters. Let us remember that in this chapter Estella begins the process of "beggaring" Pip which she carries on until the end of the novel. She is proud, beautiful, contemptuous and somebody who delights in taunting Pip because of his low birth and common nature. Consider the following quote as an example of this:

She put the mug down on the stones of the yard, and gave me the bread and meat without looking at me, as insolently as if I were a dog in disgrace. I was so humiliated, hurt, spurned, offended, angry, sorry--I cannot hit upon the right name for the smart--God knows what its name was--that tears started to my eyes.

The moment that Pip begins to cry he records that Estella looked at his tears with "delight in having been the cause of them." Estella is presented as a cruel female who delights in torturing males and gaining power over them. This is of course in massive contrast to Pip and his innocence, naivety and basic goodness.

However, there is one massive similarity between Pip and Estella that unites them together in spite of these differences. Both are (supposedly) orphans who are brought up by other parental figures in their life and are abused and damaged as a result. Just as Miss Havisham brings Estella up to not be able to love, so Pip is corrupted by the great expectations that his substitute father, Magwitch, supplies him with. Both Miss Havisham and Magwitch "adopt" Estella and Pip for their own selfish reasons.