A simplest example is a curve $C$, then $Bir(C)\cong Aut(C)$

A careful note that $Bir(X)$ is not a group scheme in general.Moreover if $X$ and $Y$ are birationl then $Bir(X)$ is not isometric with $Bir(Y)$

So we need to add some condition on $X$ in the sense of Minimal Model Program such that in scheme group theoretic sense $Bir(X)$ good behave since we have luck of well defined multiplication rule. In fact is $X$ be a minimal model with terminal singularities it is a theorem that $Bir(X)$ is a group scheme as an Abelian variety. In general $$Aut^o(X)\cong Bir^o(X)$$

where $Aut^o$ is the identity connected component. See this paper

Let $X$ be a projective variety then we have the following isomorphism from MMP

$$Aut(X_{min})\cong Bir(X_{min})$$ where $X_{min}$ is the minimal model of projective variety $X$

For any surface of non-negative Kodaira dimension, we have
$$Aut(X) ⊂ Bir(X) \cong Bir(X_{min}) \cong Aut(X_{min})$$

If $X$ be a Fano variety of $dim\geq 5$ and Picard group is generated by anticanonical divisor of the variety $X$, then it is conjecture that $Aut(X)\cong Bir(X)$

Moreover for moveable log pair $(X,M_X)$ which is log Calabi-Yau pair, with at worst canonical singularities then $Aut(X)\cong Bir(X)$