It's worth noting, first, that the phrase audire missam goes back at least as far as St. Ambrose (AD c. 339-397).
Moneo etiam ut qui juxta Ecclesiam est, et occurrere potest, quotidie audiat missam. (Sermon XXV, n. 5)
The Dictionary of Medieval Latin from British Sources (DMLBS) lists a separate ecclesiastical meaning in 5:
5 (eccl.) to hear: a (mass); b (confession).
- a 1199 cum ipsi post missam ~itam equitarent versus forestam CurR I 91; venit rumor ‥ quod cancellarius vellet transire per S. Ædmundum apud nos missam in crastino ~iturus Brakelond 135; s1214 cum ‥ in missis ~iendis fuissent occupati W. Newb. Cant. 516; s1271 in ecclesia Fratrum Minorum ‥ missarum solempnia ~ientes Leg. Ant. Lond. 134; [Henricus VII] attente ~iebat quotidie binas ternasve missas quas vocamus P. Verg. XXVI 617.
- b 1214 confessiones mulierum ~iantur in propatulo quantum ad visum hominum, non quantum ad auditum Conc. Syn. 32; 1221 quod in nullo jura dicte ecclesie, vel ~iendo confessionem parochianorum ‥ vel divina eis celebrando ‥ diminuet Ch. Sal. 116; 1228 sacerdos de A. ~it confessiones iiij villarum Feod. Durh. 228; s1300 archiepiscopus ‥ octo Predicatoribus et octo Minoribus tantum dedit licenciam predicandi et confessiones ~iendi cum per rectores vel sacerdotes ecclesiarum ad hoc officium sint vocati Ann. Worc. 546; c1520 potestates et facultates ~iendi et faciendi confessiones Conc. Scot. I cclxxiii.
This doesn't quite answer your question, since (as cmw notes in the comments) it's possible this is just an extended sense of the classically attested meaning "I receive instruction from" someone. It is worth noting, in this regard, that English "attend" has the same original meaning: "to stretch toward" something. In fact, L&S gives "attend" as one of the meanings of audio, though it clearly means the earlier sense of "pay attention to."
In short: perhaps there isn't much difference after all in English "attend Mass" and Latin "audire missam."