One advantage is that sending Romeo a letter, rather than delivering the news himself, keeps Friar Laurence from being seen as directly associated with Romeo. It may have been considered a criminal offense for family members or friends to directly contact exiled society members. Hence, if Friar Laurence sends someone else to Mantua instead of himself, he keeps his association with Romeo a secret.
However, there are far more disadvantages to Friar Laurence's decision to inform Romeo of such serious news via letter. One disadvantage is length of time. In sending a letter, Friar Laurence runs the risk of the letter not arriving in time. A second disadvantage is that Friar Laurence runs the risk of the letter not actually arriving at all, which is exactly what happened. Friar Laurence gave the letter to Friar John who felt he needed a traveling companion to go with him to Mantua. Friar John went in search of another clergyman who was visiting the sick, and wound up being quarantined with his friend. As Friar John states, after finding his friend,
The searchers of the town,
Suspecting that we both were in a house
Where the infectious pestilence did reign,
Seal'd up the doors, and would not let us forth." (Act 5, Scene 2)
Since Friar John never made it to Mantua, Romeo never received the letter. As a result, as soon as heard of Juliet's death, he believed her death was real and made a plan to kill himself. Hence, Friar Laurence's decision to tell Romeo the plan via letter, rather than traveling himself to Mantua to tell Romeo the news face to face, jeopardized not only the plan but both Romeo's and Juliet's lives.