It's not so much the size of the space, but how tightly the letters of each word are kerned. In a proportional font, all the letters in a word are more tightly packed together, making each word appear as a more separated object. This in turn makes spaces between words easier to see.
And depending on the typeface/typesetter, sentences might also be set with more space after them than is normally between words, though it's quite unlikely to be twice as much. (TeX will do this, for example.)
