Yes and no.
Yes, the way the ecig industry and users define them the two are translated to mean the same thing.
No, because when listed as a percent the amount isn't really accurate. For example, 10mg
juice means there are 10mg of nicotine per ml of liquid. However, the actual weight of the ejuice varies depending on the PG/VG mixture and flavorings added to it. Different liquids have different densities, and thus different amounts of mass and weight for the same volume.
The calculation of 1ml = 1mg is only accurate when dealing with distilled fresh water. If you're talking about gasoline then 1ml = .75mg. If you're talking about seawater then 1ml = 1.02mg.
PG = 1.04g per ml
VG = 1.26g per ml
So while your nicotine concentration of your juice may be 10mg per ml that doesn't necessarily = 10% nicotine. If you were mixing a batch of 100ml of pure VG juice then 10% nicotine would be 12.6mg per ml. But if you go to a shop and ask for 10% nicotine the bottle they give you will only be 10mg per ml.