The Stone-Čech compactification of $\mathbb{R}$ is not its one-point compactification. The former is the largest compactification of a space, while the latter, if it exists, is the smallest compactification, and in general there will be many compactifications in between. $\mathbb{R}$ has, for example, a two-point compactification, namely $[0, 1]$.
The C*-algebraic construction of the one-point compactification is not $C_b(\mathbb{R})$ but the subalgebra
$$\{ f \in C_b(\mathbb{R}) | \lim_{x \to \infty} f(x) = \lim_{x \to -\infty} f(x) \}$$
and taking the limit is the character corresponding to the extra point. (In general, if $X$ is locally compact Hausdorff, then we can construct its one-point compactification by adjoining a unit to $C_0(X)$.)
Requiring only that the limits exist but not that they agree gives the two-point compactification. More generally, points in the Stone-Čech compactification can be described in terms of ultralimits.