There is a distinction between the two, and the office of acolyte is reserved to men alone. Canon 230 §1 says: "Lay men whose age and talents meet the requirements prescribed by decree of the bishops’ conference can be given the stable ministry of lector and of acolyte, through the prescribed liturgical rite . . . "
The office of altar server was created around a thousand years ago so there would be a group of people who could stand in for acolytes when none were available. Today that is the case in most parishes, and therefore the position of altar server has come to predominate at most Masses.
However, because the position of altar server was created by the Church, the issue of who can serve as its occupants is determined by the Church. The office is not sacramental and so need not be reserved to males. Currently, the Church allows the use of female altar servers: "Where the needs of the Church require and ministers are not available, lay people, even though they are not lectors or acolytes, can supply certain of their functions, that is, exercise the ministry of the word, preside over liturgical prayers" (CIC 230 §3).
The position of altar server involves much less responsibility than the position of acolyte. Acolytes had many duties that have now been broken up and distributed to different people—for example, carrying the cross during the opening processional used to be performed by the acolyte, but now the crucifer fulfills that role. Today, nearly all that remains for altar servers is to hand the priests the unconsecrated elements and the cruets and to help him wash his hands.