So I now have Miracle Mask and badly want to play it. But I also have The Last Spector, which I haven't played yet.
Would I miss out on much story, characters, etc etc if I skip The Last Spector for now and jump right into Miracle Mask?
Yes. Don't skip it if you plan on playing it anyway. The games work well in the intended order. Only skip Layton games if you don't plan on playing them at all or don't own them and don't plan on buying them.