Is there any particular reason some priests will sing through [the Mass] and others do not? Is there a specific significance to the singing?
Pope St. Pius X probably said it best when he instructed us: “Do not merely sing during Mass, but rather sing the Mass.”
The music which we sing at Mass is not decoration for the Mass. We don’t throw together a couple of hymns to make the experience more solemn or joyful or entertaining or beautiful. True liturgical music is not a mere aesthetic, and it ought to conform to the sacred texts of the Mass which are given to us by the Church.
In the same way, the singing of the priest is not merely an aesthetic affectation. St. Augustine is credited with the phrase, “Quis cantat, bis orat” (He who sings, prays twice). In the Mass, it is the priest who prays with and on behalf of the people, his prayer always directed to the Triune God. If we place this prayer in the context of what St. Augustine tells us about the nature of singing, it is logical to presume that the chanted (or sung) prayer of the priest is quite appropriate, as we can do nothing more sublime and profound than worshiping God in the Holy Sacrifice of the Mass.
Now, why do some priests sing and others do not? Well, since the promulgation of the current liturgical books in 1969, the Church has granted much more latitude and room for the personal preference of the priest with regard to when something is sung. There are still parts of the Mass that the Church requires to be sung (such as the Alleluia before the Gospel), but there is a great deal more room for not singing certain things. With the coming of the new translation of the Roman Missal this Advent, there has been a much greater emphasis on singing the Mass. But for some, it still remains a matter of the personal preference of the priest.
We also have to take into account that priests are people too. And not everyone has been blessed by God with a talent for singing. So, some priests prefer not to do so because their singing may cause a drop in Mass attendance!
Without getting into all the various documents on liturgical music (and there are many!), and without the time (or blog space) to give a more comprehensive answer, singing the Mass is significant because it takes the Mass out of the realm of the ordinary. The Mass itself, as a mystical event, unites Heaven and earth, and the Sacred Mysteries contained therein take place outside of time and space. The manner in which Mass is said, therefore, ought to reflect that very ideal—that the Mass is something out of the ordinary and unlike what is commonplace and mundane. We sing the Mass because it elevates our minds and hearts to a different plane of consciousness and makes us more acutely aware that something truly extraordinary is taking place—God truly enters into our midst, and it is right and fitting to Sing to the Lord!
For a little extra reading, here's an interesting and informative post on the subject from The Authentic Update.