Also to clarify, we will be placing mint julep junipers directly behind either one to form a "wall"

Both are tough, hardy plants. Nannyberry probably has a more open "natural" look and is somewhat more upright than straight Cornus mas (the cultivar 'Golden Glory' has an upright habit). In my area, nannyberry is susceptible to a leaf spot disease which isn't a serious problem but detracts from the appearance of the foliage. Both can have nice fall color. Cornus mas offers early spring color which is nice after the long Michigan winter. Fruiting shouldn't be much of a problem with either - birds usually pick off most of the fruit.

Nannyberry (Viburnum lentago) seems to grow faster and more upright than the other viburnums. Nice white flowers, dark berries, and very good fall color. Very hardy. Sometimes it gets a powdery mildew on the leaves. I don't know about Cornus mas.

lentago is quite variable here. I'd definitely select prunifolium over lentago. Much more reliable fall color. Nicer branching structure too.

Cornus mas looks goofy as a shrub to me. As a tree...very nice. As it matures the bark is pretty cool looking.

Good point by nurseryman - powdery mildew can be a problem (as the leaf spot is) on nannyberry.
And I concur with whaas that prunifolium is probably a better choice than lentago overall. There is a lentago x prunifolium hybrid called 'Prairie Classic' that has good qualities (compact habit, nice fruit display, reliable fall color) - might be hard to find, though.
Also agree with whaas on the attractive bark on mature C. mas.

Ditto what Arbordave said about the V. lentago - I have about 5 decent-sized shrubs, purchased as seedlings in 2008. They have just started to flower and fruit this year and last. There is variation in the seedlings, but some of them do seem to get ratty leaves. I would purchase a cultivar that is resistant to foliar diseases.

Also, if you are in a region where the Viburnum beetle is active, this could be a problem with these species.

The birds will eat the berries before you even notice - so the mess is not an issue. Supposedly the berries are edible for humans too!