Which is?
Not trying to be obtuse here, but while I understand the difference between the vows of members of religious orders and the promises of diocesan clergy/members of religious societies must be more than just semantics, I’m not sure from a practical standpoint what that difference is. Is it a difference in the level of red tape needed to be dispensed from them? A difference in the implications of/punishments for breaking them?
YoungTradCath explained the chasity/celibacy distinction quite well, but I think your question has more to do with “What’s the difference between a promise and a vow?”.
A promise is way easier to break out of. Essentially, the only blocking issue would be for the church to laticize one (i.e. bring you back to the state of the laity, not clergy).
A vow is a commitment wherein one can only break it through the pope’s intervention.

There are generally three types: simple (yearly) vows (generally taken in the novitate years), simple (permanent) vows (generally taken by congregations, apostolic communities, etc), and solemn (permanent) vows (taken by religious orders (dominicans,franciscans,etc).
While Canon Law makes the distinctions between simple and solemn, the short end of the vows vs promise debate would be that, were a priest to attempt to marry, the secular priest would have an illicit but valid marriage (and could, following the proper channels, remove himself to the laity and be able to legally (and validly) marry), while the religious priest can never marry validly.
…It can be a little confusing, and this is a rather simplistic overview. If you have more questions about it, asking one of the Brothers on the forums would probably give you a fuller answer.