That means had he survived that winter, he would have obviously focused on Italy, and probably would have dislodged the duchies of Benevento and Spoletto.
I'm not sure this is at all obvious- while by 602 the Balkan situation was largely under control, I suspect there'd be continuing campaigns in the area for several years, maybe even up until the death of Maurice. His successor would then in all likelihood be facing renewed Iranian problems.
Anyway- continued Constantinoplitan presence in central Italy means that, yes, the Bishop of Rome won't be anything like so influential, and the institutional Catholic Church as we know it won't exist. Maybe many more national churches in Western Europe under the thumb of Frankish and English kings. The Bishop of Rome may well end up no more influential than any other (non Constantinopolitan) Patriarch.
Italy will likely remain something of an imperial backwater- the priority will always be on the East, unless Iran/the Caliphate is utterly fragmented somehow, and a large and powerful Northern European state emerges to threaten the Mediterranean core.
Finally, I think in linguistic terms Italy (outside of Greek speaking Calabria) stays Latin, but knowledge of Greek in Italy and Latin in Constantinople is somewhat higher than OTL.