While not necessarily an authoritative source, I think the Marvel Wikia entry for Secret Avengers (Black Ops Group) summarizes it decently:
When Steve Rogers took over Norman Osborn's role as top cop, he chose not to continue H.A.M.M.E.R. or to reestablish S.H.I.E.L.D. Instead, he decided to put the world's security into the hands of a few Avenger teams. One of these teams became the Secret Avengers, and he took direct leadership of it with Sharon Carter assisting him with tactical operations. The purpose of this team was to covertly clean up the messes left by the Osborn administration. To this end, Steve personally recruited a team that would allow him to carry out surgical strikes in any given circumstance.
The formation of this group, shouldn't be confused with the Secret Avengers formed as part of the Civil War story arc, however, and it's never stated anywhere that I'm aware of, Rogers may have borrowed his former group name under which to carry out these black ops missions.
Though I'm unaware of any canon that's explicitly stated such, it's my opinion that there are two possible reasons why the group operates in relative secrecy:
Concern over disagreement and undermining of some of the groups operations by other, more conservative heroes [disagreements in larger groups being more common]. Roger's formation was based on trust and ability. This limits risk of mission endangerment... particularly as some of their missions may be the sort of thing other Avengers could find objectionable.
Smaller teams lend to greater cohesion and greater capacity for surgical strikes/operations. Efficiency.
Of course, though you added it, perhaps, in jest ... jealousy and the negative impact on relationships with other Avengers should an overt team within the team occur may have played some part too. Past experiences have occasionally shown such to lead to undesirable outcomes.