Are two definitions of affine varieties totally parallel?

Question

Vakil defines the affine variety differently from Hartshorne, which is the spectrum of the coordinate ring of a affine variety in the sense of Hartshorne. I wonder if they share all the properties (like the theorems, definitions and so on)? I know there is a fully faithful functor from varieties (in the sense of Hartshorne) to the schemes, which sends an affine variety (in the sense of Hartshorne) to an affine variety (in the sense of Vakil), but does that guarantee a theorem of varieties in the sense of Hartshorne is always true when we realise the varieties in the sense of Vakil for the same statement of the same theorem?

Answer 1

They're not literally equivalent (Vakil's varieties have nonclosed points, those of chapter 1 in Hartshorne don't), but anything which can be talked about in categorical terms which is true in one setting will be true in the other (*once you make the appropriate addition/deletion of hypotheses). Fortunately, this is most properties we care about in algebraic geometry. Many/most proof techniques from one setting can be used in the other to prove the same results, too.

*: To be clear on the hypotheses, after applying the scheme-ification functor, Hartshorne's varieties are irreducible reduced separated schemes of finite type over an algebraically closed field, while Vakil's are just separated schemes of finite type over a field.