In a guesstimated 99,9% of cases, names should **only** go into "Name" field of Potlatch 2. The "Building Name" field is intended for the comparatively rare case of names that are part of a postal address.
address. As the building in your example has a house number, it is probably not one of these special cases.
Within the OSM database, the "Name" field will fill in the [name][1] key, whereas the "Building Name" field fills in [addr:housename][2].
When you are interested in subtle differences in meaning, it often helps to look into the "advanced" tab to find out which tags are hidden behind the user-friendly facade.
[1]: http://wiki.openstreetmap.org/wiki/Key:name
[2]: http://wiki.openstreetmap.org/wiki/Key:addr:housename