There are several disadvantages with the use of the term "pericope" for Synoptic-Gospel units of text. The first is that it is a jargon-word and functions mainly to make scholars feel superior to everyone else. It's "in" language. If you talk about a Biblical "passage", you are not one of us! You are probably doing a horrid Bible Study with your horrid, non-academic friends, and you have no idea about how to study the Bible academically. If you were an academic, you'd talk about pericopes!
The second disadvantage is that no one knows how to do the plural of "pericope". It's pericopae if one is being truly clever and "in". But some scholars worry that this is a step too far. It's just too ponsey. So they go for "pericopes". And that looks even more ungainly.
The third great problem with the word is that it cannot survive in the age of auto-correct. MS Word knows that there is no such word as "pericope". Surely, you must be intending to write "periscope". And it will change it for you. Without even asking you. It's a free chuckle every time I see "periscopes" in a student paper. And lest I become too puffed up, I have found it more than once in my own hand-outs, where auto-correct has done its worst.
In any case, what was ever wrong about talking about "passages" in the first place? The word "pericope" is in any case a legacy of the kind of form-critical approach to the Gospels that we have all long-since abandoned. Austin Farrer so disliked that kind of atomistic approach that he used to talk about "paragraph criticism", noting that he was quite happy to look at isolated paragraphs but that, in the end, it was the Gospel as a whole that rewarded careful, critical study.