What does the Church say about the difference between me saying Prayer before meal and a Priest or Deacon doing it?
The only difference is that a priest gives something a priestly blessing. Deacon are permitted to bless certain things while employing the sign of the Cross like a priest. Food is amongst those things a deacon is permitted to bless. A layperson May bless himself but may not employ the sign of the Cross over the meal they are about to partake in.
Generally speaking, it is the most senior ecclesiastical person who says the blessings over a meal. This is the norm in traditional houses of formation like Benedictine Monasteries.
Nevertheless if a monk or nun arrives late to a common meal, they say grace themselves before sitting down to eat.
In private a priest, deacon or layperson may employ the aforementioned blessing over their meals. I have been using this particular blessing myself for many years now. It is only natural to ask to highest ranking ecclesiastic individual whether a bishop, priest or deacon, the privilege to bless the food we are about to consume.
The following two prayers are designated for laity and are derived from the full prayers in the Breviary that date back to at least the 9th century but are likely of a considerably earlier origin as they are similar to prayers found during the 4th century. These prayers are to be said aloud before after after each meal and are to be led by the head of the table. - Prayers for Meals
Benedic, Domine, nos et haec tua dona quae de tua largitate sumus sumpturi, per Christum Dominum nostrum. Amen.
And do not forget thanksgiving after your meal:
Agimus tibi gratias, omnipotens Deus, pro universis beneficiis tuis, qui vivis et regnas in saecula saeculorum. Fidelium animae, per misericordiam Dei, requiescant in pace. Amen.
By the way: Bon Appetite!