I heard a formal group law is a formal group with chosen coordinate. But I cannot grasp this meaning.
My understainding; （1-dimmensional）formal group and formal group law are both element of formal power series with two variables, which satisfies some conditions（like group axiom）. For example, $X＋Y∈K[X,Y]$ is formal additive group law.
Could you tell me the difference between formal group and formal group law with examples?
Thank you so much for your help.
For Silverman's purposes, I think it's best if you just thought of this as a notational choice and moved on quickly to something with more content.
In any case, Silverman is clearly thinking of formal groups as objects in some category and formal group laws as a structure internal to those objects, so he uses $\mathcal{F}$ for the formal group (object in a category) and $F(x, y)$ for the formal group law (a power series). However, for his purposes in Section IV.2 there is no additional structure to $\mathcal{F}$ beyond $F(x, y)$, so you're asking why there are two symbols.
Well, you could just use $F(x, y)$, but if you did things would get awkward. For instance, a homomorphism between formal groups would then be a function $f \colon F(x, y) \to G(x, y)$ such that $G(f(x), f(y)) = f(F(x, y))$--the notation specifying the domain and codomain of $f$ is pretty misleading.