There are several layers of distinction here. Your wife (and many others) are mainly aware of a difference in vocal style; this is simply a difference between the style used in "classical" "art" music and other modern genres.* It might sound less like opera, by the way, if you listen to a recording that is grounded in historic performance practice; much of the bel canto technique that we associate with opera arose in the 19th century, well after Bach's time. In this recording with the Freiburg Baroque Orchestra and St. Thomas Boys' Choir, the vocal style would still not be mistaken for modern pop, but lacks the wide vibrato and powerful projection that people associate with Bugs Bunny in a Brunhilde costume:
(Note, opera itself is much older than modern operatic technique; opera was big business—show business—long before Bach. The technique of the 17th century is a matter of much research and conjecture, but it certainly wasn't Wagnerian; it's not a matter of "historic baroque style vs operatic style.")
As for what makes a mass, as a type of work, different from an opera, let's answer a different question first: what makes an oratorio different from an opera? The most famous oratorio is easily Handel's Messiah, but it's a bit of an atypical example. Take a different Handel oratorio then, Israel in Egypt. Like an opera, it has a storyline. It has characters, sung by various soloists. It has a chorus for the "big numbers." What it lacks is staging—no costumes, no scenery, no walking around the stage.
So what makes a mass different than an oratorio? This is a bit like asking "what makes a church service different than a Broadway show." A mass is a religious service. Its format has been in place for centuries (with slight tweaks), with components still referred to in Latin (and Greek)—the Kyrie, the Gloria, the Credo. A musical composition that calls itself a mass simply sets these liturgical texts to music.
* This is because, when people encounter an unfamiliar culture, they're at first highly aware of how it differs (as a whole) from familiar cultures, and not aware of differences within the culture. For instance, to Western listeners with little experience of Indian music, it's hard to notice the differences between different ragas and talas; it all "sounds the same" because they're too aware of the characteristics that are common to all the Indian examples but not to Western music: drone, certain timbres of plucked strings, instrumentation, etc.