What a great comparison. We have sailed twice with Viking and the last three with Regent. With Regent, it really depends on which ship you book. We loved Explorer, but would not sail again on Navigator.
If you are a couple and like to dine alone, Regent is much more preferable. With Viking you are very close to the next table, while Regent is more similar to a nice restaurant. The food, in our opinion, is a significant notch above Viking.
One huge negative about Regent is their Air department. We had so many issues with them on our recent cruise, despite paying for custom air, that we have not booked another sailing. That is just our experience and others may feel differently.