The definition for $C_0(\Omega)$ is continuous functions that vanish at the boundary of $\Omega$. When $\Omega = \mathbb{R}$ the boundary is an empty set thus we have $C(\mathbb{R}) = C_0(\mathbb{R})$.
However some books define $C_0(\mathbb{R})$ to be continuous functions that vanish at infinity, then non zero constant functions will not be in $C_0(\mathbb{R})$ but they will be in $C(\mathbb{R})$.
Are the two $C_0(\mathbb{R})$ different?
That's not the definition of $C_0$ I have used my whole life. The way I understand and use $C_0(\Omega)$ (and they way I have always seen it in the literature) is $$ C_0(\Omega)=\{f\in C(\Omega):\ \forall\varepsilon>0,\ \exists K\subset\Omega\ \text{ compact, with } |f(t)|<\varepsilon,\ t\in \Omega\setminus K\}. $$ That is, functions that are arbitrarily small outside of a compact set. In other words, $C_0(\Omega)$ is the norm-closure of the set of continuous functions of compact support.
In particular, a $C_0$-function may be nonzero everywhere.