All baptized persons may bless, even non-Catholics. But not all blessings may be given by all people.
As a vague rule, blessings involve things and persons over which one has been granted spiritual authority by God. For this reason, parents can appropriately bless their children, persons their homes, and those with appropriate Holy Orders can bless water, which furthermore serves to consecrate it: it becomes holy water. For a lay person to attempt to give certain blessings would not be in the proper order of things. People have different vocations: some acts are only appropriate to those who have been called by God into the priesthood, and very inappropriate to others.
Specific blessings differ in form in much the same way that specific prayers differ in form. As the solemnity of the act increases, so too does its formality. Very important blessings will have certain strict forms.
So one difference is that priestly blessings tend to be more formalized. The other differences, already mentioned, are that particular priestly blessings are restricted to priests, and that some such blessings serve to consecrate.
From the Catechism:
1668 Sacramentals are instituted for the sanctification of certain ministries of the Church, certain states of life, a great variety of circumstances in Christian life, and the use of many things helpful to man. In accordance with bishops' pastoral decisions, they can also respond to the needs, culture, and special history of the Christian people of a particular region or time. They always include a prayer, often accompanied by a specific sign, such as the laying on of hands, the sign of the cross, or the sprinkling of holy water (which recalls Baptism).
1669 Sacramentals derive from the baptismal priesthood: every baptized person is called to be a "blessing," and to bless. Hence lay people may preside at certain blessings; the more a blessing concerns ecclesial and sacramental life, the more is its administration reserved to the ordained ministry (bishops, priests, or deacons).
1670 Sacramentals do not confer the grace of the Holy Spirit in the way that the sacraments do, but by the Church's prayer, they prepare us to receive grace and dispose us to cooperate with it. "For well-disposed members of the faithful, the liturgy of the sacraments and sacramentals sanctifies almost every event of their lives with the divine grace which flows from the Paschal mystery of the Passion, Death, and Resurrection of Christ. From this source all sacraments and sacramentals draw their power. There is scarcely any proper use of material things which cannot be thus directed toward the sanctification of men and the praise of God."
1672 Certain blessings have a lasting importance because they consecrate persons to God, or reserve objects and places for liturgical use.
(A blessing is a sacramental. A sacramental is not the same as a sacrament.)
Attempting to perform certain blessings as a lay person is a bad idea. For example, this canon lawyer's blog is emphatically opposed to lay ministers attempting to confer a blessing using the Holy Eucharist:
Blessing the faithful with the Most August Sacrament is expressly reserved to the ordained. Lay persons may not confer any blessings with the Host (Eucharistic worship outside of Mass nn. 91, 97-99, and 1983 CIC 1168). This practice should therefore be immediately halted wherever it has cropped up.
Generally, things that you see only priests doing are things that only priests should do.
(And if you're unsure, it's best to find out - ask the priest.)
A couple concrete examples of formulaic differences from http://forum.catholic.org/viewtopic.php?t=14564 (all from the first post)
In the blessing of a table or meal: when the blessing prayer is said,
the classic prayer, "Bless us, O Lord, and these thy gifts...." the
layman who does it makes the sign of the Cross on himself, as usual,
while the deacon or priest makes the sign over the food.
And for the priestly version of blessings:
In the name of the Father, and of the Son, and of the Holy Spirit.
Grace to you and peace from God the Father, and from our Lord Jesus
Response: And with your spirit (Et cum spiritu tuo.
Versus the lay version:
The grace of our Lord Jesus Christ be with us all.