They are different fonts from different font families, not members of the same family, and Helvetica is far, far more common than Helvetica Neue.
The correct hierarchy of typographic terminology, as I learned it, is broken down into the four Fs.
FOUNDRY: The designer and/or manufacturer of sets of type. Berthold Direct Corporation is a type foundry.
FAMILY: Related typefaces from the same foundry. Berthold Garamond is a type family.
FACE: The style of the type (regular, medium, bold, etc.). Berthold Garamond regular is one typeface while Berthold Garamond italic is a different typeface.
FONT: A typeface at a specific size. Berthold Garamond italic, 10 point is one font while Berthold Garamond italic, 12 point is a different font.