What are some similarities between Oliver Twist and Pip (from Great Expectations)?

The protagonists of Great Expectations (Pip) and Oliver Twist (Oliver) have several similarities. The most obvious is that both are orphans. Chapter one of Great Expectations explains that

"I [Pip] never saw my father or my mother, and never saw any likeness of either of them . . . "

Pip was orphaned very young and has no memory of his parents's appearances. At the start of the story, he is living with his sister and brother-in-law. Unfortunately, his sister is not known for being a kind woman:

"My sister, Mrs. Joe Gargery, was more than twenty years older than I, and had established a great reputation with herself and the neighbors because she had brought me up “by hand.” . . . knowing her to have a hard and heavy hand, and to be much in the habit of laying it upon her husband as well as upon me, I supposed that Joe Gargery and I were both brought up by hand" (ch. 2).

Pip's older sister is known for her "heavy hand," or the physical punishments that she uses to motivate (or perhaps abuse) both Pip and her husband. Pip shows his fear of his sister (his guardian) at the start of the story. His living arrangements are very uncomfortable.
Oliver also is an orphan. His mother gave birth to him in a workhouse. After she was handed her son, she died:

"The surgeon deposited it in her arms. She imprinted her cold white lips passionately on its forehead; passed her hands over her face; gazed wildly round; shuddered; fell back—and died" (ch. 1).

Oliver has no knowledge of who his father is, and he is given to the parish (or church) to care for his infant needs. However, the parish leaders care little for his well-being and try to get rid of their responsibility of him as soon as he is old enough.

"For the next eight or ten months, Oliver was the victim of a systematic course of treachery and deception. He was brought up by hand. The hungry and destitute situation of the infant orphan was duly reported by the workhouse authorities to the parish authorities."

Like Pip, Oliver was "brought up by hand." Dickens shows how both Pip's and Oliver's early living situations were uncomfortable. Like Pip, Oliver also grows to fear many of his early guardians. For instance, Mrs. Mann, his guardian in chapter 2, locks up children who complain that they are hungry:

"it was his [Oliver's] ninth birthday; and he was keeping it in the coal-cellar with a select party of two other young gentleman, who, after participating with him in a sound thrashing, had been locked up for atrociously presuming to be hungry . . ."

As for differences, I'd argue that Oliver shows more bravery than Pip does in the midst of great and fearful challenges. For instance, when he and the other boys are starving in the workhouse, Oliver is selected by lots (or at random) to ask for more food. At mealtime, Oliver is commissioned by his friends to approach the master and ask for more food:

"'Please, sir, I want some more.'"

Oliver could have argued, complained, or refused to ask the master for more food. He knows that he will get reprimanded for asking for more than their allotted serving. Yet he takes the blame and punishment. He willingly stands up to his master on behalf of himself and the other boys.
Later, we see Oliver's bravery when he decides to run away from his living arrangements with Mr. Sowerberry (the undertaker). When he is mistreated, he bravely decides to run away. He takes the initiative rather than allowing others to command his future. He tells his friend:

'You musn't say you saw me, Dick,' said Oliver. 'I am running away. They beat and ill-use me, Dick; and I am going to seek my fortune, some long way off. I don't know where . . .' (ch. 7).

Pip, however, shows less bravery. He does not take initiative but generally allows others to command his future. This is seen in chapter one when he meets a criminal. The criminal demands:

"“You bring me, to-morrow morning early, that file and them wittles. You bring the lot to me, at that old Battery over yonder. You do it, and you never dare to say a word or dare to make a sign concerning your having seen such a person as me, or any person sumever, and you shall be let to live. You fail, or you go from my words in any partickler, no matter how small it is, and your heart and your liver shall be tore out, roasted, and ate."

This criminal threatens Pip's life, and he obeys the man's instructions (probably because he is afraid of him):

"I said that I would get him the file, and I would get him what broken bits of food I could, and I would come to him at the Battery, early in the morning."

He does help the criminal, and he doesn't try to get him in trouble. He does not try to escape his misfortunes on his own (as Oliver does)—instead, he waits for others to assist him. This is seen later in the story when Pip meets the same criminal when he is a young adult. Pip recognizes the man immediately, and a few minutes into their conversation, the criminal declares:

"'Yes, Pip, dear boy, I've made a gentleman on you! It's me wot has done it! I swore that time, sure as ever I earned a guinea, that guinea should go to you.'"

The criminal is giving Pip money to better his future. Pip relies on others, rather than his own abilities, to change his future. Pip, rather than being immediately thankful, is upset by this revelation:

"The abhorrence in which I held the man, the dread I had of him, the repugnance with which I shrank from him, could not have been exceeded if he had been some terrible beast."

Oliver is also helped by criminals in Oliver Twist. Oliver, however, is thankful for the food and shelter that Fagin gives him. Even after he determines that Fagin is a criminal, he generally seems thankful when anyone, even criminals, helps him to meet his physical needs. Still, Oliver tries to escape the criminal's company as soon as possible; he does not want to become a criminal too.