Practically, this means that a friar's/monk's/nun's ministry and life in the community is directed by a fellow friar/monk/nun who is elected to authority by the community. For DP's, their ministry and life in the diocese is subject to the bishop. Now, all religious orders within a diocese are subject to the bishop in so far as that bishop must approve any religious ministry in his diocese. Bishops have no authority over the internal workings of a community. So, if a priory or monastery elects as prior/abbot someone the bishop doesn't like, he is not empowered to dispose of that election. He can revoke the faculties of the priests in the house, or fire any offending religious who works for the diocese. But he cannot step into the internal affairs of religious.
There are other prominent differences between RP's and DP's. One big difference is the taking of religious vows. RP's are made religious priests by making solemn vows regarding poverty, chastity, and obedience. DP's do not make religious vows. At ordination, all priests promise chastity and obedience to an "ordinary" superior. For religious priests at ordination, we make these promises to both our immediate superior and the bishop. DP's do not take a vow of poverty b/c they are considered "self-employed" by the IRS. RP's usually have access to community cars, funds, medical care, room and board, and other essentials for daily living. DP's provide most of these for themselves as "employees" of the diocese. In practical terms, the vow of poverty is about not owning anything in one's own name. RP's cannot own a car. DP can. Same goes for houses, boats, etc.
Another big difference is spirituality. RP's often belong to order's with long traditions in certain kinds of spirituality. Think: Ignatian Exercises for the Jebbies. Or the spirituality of "prayer and work'" for the Benedictines. Dominicans consider our daily lives lived according to the constitutions to be our spirituality. There is a spirituality for DP's. The big difference is that DP's rarely live in community. There prinicple spirituality revolves around their ministries in direct service to their parish.
This brings up several other differences rooted in ministry:
DP's work within the limits of their dioceses (there are exception for academics and others)
RP's can work anywhere in the world where their order has a house.
DP's usually work in parishes or ministries that directly serve the laity (exceptions: ditto)
RP's often work in universities, hospitals, secular jobs, etc. where the focus is not necessarily on serving the parochial laity directly (exceptions: many RP's serve parishes)
DP's have fewer opportunities to "switch ministries" b/c their immediate superior (the bishop) has responsibility for ministries only within his diocese and parishes need priests
RP's have much more flexibility in this regard b/c their assignments are made by superiors who have responsibilities beyond a diocesan border (e.g. yours truly assigned to Rome rather than a university in my province)
DP's have fewer opporunities for advanced study b/c of the pressing needs of their dioceses
RP's are usually encouraged to pursue advanced study if there is need
DP's have a more flexible daily schedule and tend to be more available for one-on-one interaction b/c they do not have community responsibilities (cooking for six or more brothers, taking care of community cars, accounts, etc.)
RP's are much more restricted by community obligations in their daily schedule and availability (communal prayer, meals, recreation time, etc.)
One interesting development since Vatican Two is the blurring of some of these lines between RP's and DP's. It is not at all uncommon now to find DP's living in small communities in urban areas where parishes are clustered together. In fact, many younger DP's are insisting on living in community as a way of maintaining accountability and fostering fraternity. At the same time, many religious, in the name of ministerial necessity, have moved out of community life and set up house in apartments or rectories to live alone. For the most part, this development was a reaction to the perceived restrictions of the community rule that some felt stifled their ministries. This trend among male religious is waning fast and in some cases actually forbidden.
One simplistic way of understanding the essential difference between RP's and DP's is to think of RP's as a bunch of guys living in a fraternity house (shudder) and DP's as guys who live by themselves as single men. This image (though deeply flawed) at least points up the day-to-day differences that emerge from the differences in living by yourself and living with your family.