There a few reasons that Darcy feels superior to Elizabeth Bennet. Some have to do with education. Some have to do with wealth. Some have to do with family connections. Let's examine one or two from each category.
While Elizabeth is plainly well educated, her education was at home and, while overseen by her mother, was in some degree up to her own choice as to how far she wished to take her educational opportunities. It could never be argued that she was uneducated or that her education was neglected but it can certainly be argued that it was not rigorous nor was it carefully overseen (her younger sisters show more of the laxness with which Mrs. Bennet later came to undertake her later daughters' education). Darcy, on the other hand, has had a gentleman's education at the best, most exclusive public (in England "public" means what Americans call "private") schools and at one of the prestigious universities. In addition to the rigor with which Darcy was educated and the absence of rigor in Elizabeth's education, there was a social disparity between men's and women's educations. While it is absolutely not true to say that women of Elizabeth's class were poorly educated, it is true to say that they were not offered formal education in higher disciplines like philosophy and science and Classical literature, which is not to say that many women like Elizabeth, and Jane Austen, did not independently read in some or all of these disciplines. Therefore, education is one reason Darcy felt superior to Elizabeth.
Looking at family connections, Darcy was the nephew of Lord (deceased) and Lady de Bourgh and Earl Fitzwilliam, Colonel Fitzwilliam's father. Darcy Senior, though not titled as the titles lay on Mrs. Darcy's side of the family, had enormous wealth and prestige of his own and ran his estate and village with great care and deeply ethical responsibility. In contrast, Elizabeth's mother married above her station as her family were of the professional working classes as her brother, Mr. Gardiner, was in business in London and her sister, Mrs. Phillips, was married to a lawyer in Meryton. While Mr. Bennet's family, though he had (had) wealth and was a gentleman, had doubtable connections, like Mr. Collins, Mrs. Bennet's side had inferior connections. For this reason, Darcy felt superior to Elizabeth though he obviously did not feel superior to her individually as the daughter of an upper class gentleman.
As to wealth, Darcy had great wealth that was all his own as the heir to his father's estate. Elizabeth, on the other hand was heir to two or three hundred pound per annum (a year) upon her marriage. Considering that Elizabeth could bring nothing of greater wealth into the family as was the custom, this was another reason for Darcy to feel superior to her, though it must be emphasized that these externalities that made Darcy feel superior did not apply to feelings of superiority toward Elizabeth herself whom he saw as an equal personally.
"It taught me to hope," said he, "as I had scarcely ever allowed myself to hope before. I knew enough of your disposition to be certain that, had you been absolutely, irrevocably decided against me, you would have acknowledged it to Lady Catherine, frankly and openly. ...
"What did you say of me, that I did not deserve? For, though your accusations were ill-founded, formed on mistaken premises, my behaviour to you at the time had merited the severest reproof. It was unpardonable. I cannot think of it without abhorrence."