H Street and NoMa really aren't that far apart -- H runs through NoMa, in fact. The new apartment buildings at the west end of H St NE (over to 3rd St.) are
technically in NoMa, which spans both sides of the tracks.
Otherwise, H St. doesn't have that many new apartments just yet. There will be plenty in coming years, since demolition permits have been issued for sites like Murry's, but for the most part it's rowhouses in varying condition. Similarly, if you eventually want a condo, there really aren't many around H or in NoMa, as all of the new development has been rental.
Although H certainly has more happening, that rules it out for the OP's purpose (to get a nice new apartment for $3000+ a month). Between MVS or NoMa, I'd say it mostly comes down to where you're going on a regular basis, as they're fairly similar areas less than a mile apart. Union Station? Green Line? I-395 or US 50? Logan Circle or Eastern Market?