Adam Bell and Clym of Clough battled the evil Sheriff of Carlisle, in many ways similar to Robin Hood. It's hard to say which story influenced the other. There is very little in the oldest Robin Hood stories about Robin Hood helping the poor, although this idea was popularised by disgruntled Hollywood script writers expelled from the USA by Macarthyism.
The "Lytle gest of Robin Hood and his meiny", the earliest known ballad has Robin Hood helping a knight who has fallen onto hard times, paying his debts to the Jewish moneylenders of York. It's hard to read a socialist message into this, a knight whose son has a gambling problem is not exactly the same as a starving serf, and also the story is somewhat anti-semitic: The Jews of York were burned alive in a pogrum, and several English ballads attempt to justify this.
Robin Hood has been used for many kinds of propaganda, for different political agendas. Sometimes he is a peasant revolutionary, sometimes a loyal patriot and monarchist, raising money to pay Richard the Lionheart's ransom. I never really liked this last take on the stories (it seems to be a later element). I have some sympathy with Prince John. He signed Magna Carta, and if he did tax the people heavily it was most likely because his brother was away spending the money on pointless wars in the middle east. It's little wonder that John was reluctant to pay off his brother's kidnappers. Shakespeare portrays King John in quite a positive light. The idea of evil Prince John seems to have come along rather later.
The idea of Robin Hood the subversive anti-hero is nearer the mark. There are popular stories about other semi-mythical bandits like Dick Turpin (historical, but mythologised and muddled up with other highwaymen. The real Dick Turpin was an ugly rapist) Ned Kelly and Billy the Kid. These characters fight the rich and powerful, there is no need for them to help the poor. The name "Robhode" appears frequently in medieval court records. It may have been a generic name, like John Doe, for a criminal whose identity was unknown. Some real historical bandits and rebels may have adopted the name deliberately to identify themselves with the mythical figure. IMO attempts to identify Robin as a real person ruin some of the best stories.
To me the most interesting aspect of the Robin Hood legends is the lingering pagan elements. The Ballad of Robin Hood and Guy of Gisbourne sounds like a simulated pagan sacrifice ritual, with an actor wreathed in leaves like the Mayday Jack in the Green or the Green man carved in old churches, and a second actor disguised in a brown horse (or maybe donkey) hide "top and tail and mane" perhaps symbolising the waxing and waning year, like Osiris and Set.
The Ballad or Robin and Gandelyne in which Robin is slain by a small child named Wrennock of Don is also interesting. It is similar to the legend of the death of Hodr at the hands of Wali in the Norse myths. Many of the Celtic deities in the Mabinogion are described as children or descendants of Don. Academics dispute whether this is actually a story about Robin Hood. IMO it is at least related. The pairing of the Robin and the Wren is interesting. Both are associated with the midwinter solstice.
There is a bawdiness about the stories too. Merry old England meant England before the puritans. The ballad of Rose red and white Lilly is sexually ambiguous. Two girls join the outlaws disguised as boys and Robin gets one of them pregnant, apparently unaware that she is a girl! The name "Maid Marion" is a sexual innuendo, because "making merry" meant making love. In traditional May day parades, the character of Maid Marion was played by a young man in drag.