Thanks for the comments. @nosr: Do you know a reference for these?

Thanks to everyone, In fact what Ralph says is true. I was knowing that these two objects do not give the same group scheme. I just wanted to be sure about author's style. Because there are many important results about affine group schemes in that book and so one may expect those results to be something about Hopf algebra structures. So any beginner in this subject like me may have confusion while studying on that book. It means that I (and any reader of that book) must pay attention to this.

To Andre: As I know there is not any classification for connected group schemes (and so for their representing Hopf-algebras). In "p-divisible groups", Tate proves an equivalence between the category of connected $p$-divisible groups and the category of divisible formal Lie groups. Maybe this may help you.

Thanks, I mean Spec$A$ by saying that $A$ is a scheme. Since the categeory of affine $k$-groups are anti-equivalent to category of $k$-Hopf algebras I say like this. The author is Waterhouse. But this statement exists in many lecture notes that I can't list them now. I think what Ralph says is true, because the the critical part of the proof is the case height=1. So I think it should be understood from the theorem that " ...... A has the form $k[X_{1}, X_{2}, ..., X_{n}] / (X_{1}^{p^{e_{1}}}, ...., X_{n}^{p^{e_{n}}})$ as a $k$-algebra (not as a $k$-Hopf algebra)".