Difference between $C_0(\mathbb{R})$ and $C(\mathbb{R})$

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?