R2-D2 is one of the few Star Wars characters to appear in all of the main saga's films, but there is a key difference in his appearance between the original trilogy to the prequels. Artoo has rocket booster attachments in the prequels, and makes good use of them, but in the original trilogy, there is no use or mention of them.
The real-world answer is that George Lucas didn't have CGI technology at his disposal in 1977 to convincingly make Artoo fly. Just navigating the little astromech droid up and down ramps was a feat of ingenuity. The technology available to Lucas for the prequels 20 years later allowed the droid to make practical use of his many attachments and take flight.
A canon text offers a plausible but vague explanation for why Artoo doesn't use his rocket boosters in the original trilogy when he clearly had them in the prequels. The Star Wars Book, released in October 2020, states that "Jet-setter R2-D2 has rocket boosters built into his legs but they are damaged at some point after the Clone Wars." That makes sense, because Artoo is known for throwing caution to the wind; he could have easily damaged the rocket boosters on any number of dangerous missions.
The period between the events of Revenge of the Sith and A New Hope was also a tumultuous time when the droid might not have been able to get the repair needed after being damaged in combat. Luke even comments in A New Hope how busted up the droid is when he tries to access the message embedded in his memory system. While that's not the most thorough explanation, it is plausible and definitively canon.
Besides his daring personality, R2-D2 is not like other astromech droids manufactured by Industrial Automation: He has more attachments than other typical R2 units. There are a couple of explanations for how Artoo became so a tricked-out. The Star Wars: Revenge of the Sith Visual Dictionary, whose canon status is dubious, states that "Anakin has tweaked the droid's processor and memory matrices, and improved his interchangeable component design by upgrading his tool kit." That seems plausible, considering Anakin used to repair droids as a hobby, and would want to personalize Artoo.
Another source that is definitively canon, the Star Wars: The Complete Visual Dictionary, states, simply, "Few [astromech droids] are as well equipped as R2-D2, who has been kitted out to serve the Royal House of Naboo and, later, the Jedi Order." This uses the in-world importance of Artoo to the Republic he served to explain his bonus features. He was a trusted and valuable asset to Jedi missions in the Clone Wars and in the original trilogy. Artoo used his rocket boosters to escape Super Battle Droids in the opening sequence of Revenge of the Sith. Those boosters certainly would have been helpful in The Empire Strikes Back, when he had to navigate the swamps of Dagobah, or in Return of the Jedi, when he needed to abandon Jabba the Hutt's doomed sail barge.