For the most part, we don't. The vast majority of applications of uniform continuity that I've ever seen use the fact that every continuous function on a compact metric space is uniformly continuous. That is, we rarely ever care about the fact that $2x$ is uniformly continuous and $x^2$ is not, because typically the only settings in which we will use uniform continuity are when we have restricted our attention to a compact subset of the domain. I'm sure some people will come and give counterexamples to this, but it is very easy to be a mathematician and not once think about the notion of uniform continuity for functions on a non-compact space.
The one notable other use of uniform continuity that I know of is the following fact: any uniformly continuous map $f:X\to Y$ between metric spaces extends to a continuous map $\bar{f}:\bar{X}\to\bar{Y}$ between their completions. This fact tends to come up particularly in functional analysis, where typically $X$ and $Y$ are normed vector spaces and $f$ is linear (actually in this case linearity means that continuity automatically implies uniform continuity, so like in the previous paragraph we don't really care about uniform continuity per se). It is also occasionally useful for showing that, say, a map $\mathbb{Q}\to\mathbb{R}$ extends continuously to a map $\mathbb{R}\to\mathbb{R}$. But for functions like the functions $\mathbb{R}\to\mathbb{R}$ you're asking about, this is of course useless, since $\mathbb{R}$ is already complete.
