Service of the complaint where personal service by a constable is impossible (because the location of the deserting spouse is unknown to the deserted spouse) is accomplished by publishing notice of the divorce case in a newspaper located in the location where the now-missing spouse was last known to reside.
The advantage of pleading "desertion" over a "no fault" divorce is that the act of deserting could warrant an unequal distribution of the assets. If successful, you have convinced a judge that your ex-spouse did not have justification for leaving, but left anyway, and that could result in the Judge awarding you property that they deserted as well.
The disadvantage is that proving desertion is somewhat more complicated than a "no fault" divorce. You must prove the elements of desertion to be divorced on those grounds, and if you fail to prove that the deserting spouse left without good reason, for example, then the Judge could deny your divorce. "No-Fault" divorce is much easier to prove because the evidentiary burden is met when one spouse simply tells the court that the marriage has been irretrievably broken down with no hope of reconciliation.