I'll try to describe it without the confusing math notation. For both continuity and uniform continuity, the game starts by choosing some usually small $\varepsilon>0$.
Continuity of $f$ means that at any given point $x$, if we get within a small enough $\delta$-neighborhood of $x$ (an open set containing all points whose distance to $x$ is less than $\delta$), then $f$ only takes values within an $\varepsilon$-neighborhood of $f(x)$. Essentially, we can force the function arbitrarily close to a particular value $f(x)$ by going close enough to $x$. What "close enough" means may depend on $x$.
Uniform continuity means that if we choose an arbitrary amount of values for $x$ under the condition that none of them are farther away from each other than a suitably small $\delta$, then their corresponding outputs under $f$ are no further away from each other than $\varepsilon$. Essentially, all intervals of length $\delta$ are mapped to within intervals of length $\varepsilon$, no matter where those $\delta$-intervals are. Or put another way, if $x$ varies by less than a suitably small $\delta$, then $f$ varies by less than $\varepsilon$.
Now to counterexamples to your hypothesis: bounded continuous functions are uniformly continuous. This is not true. Consider for instance $f:(0,\infty)\to\mathbb R,~f(x)=\cos\left(\frac1x\right)$. This is clearly bounded and continuous, but looking at a graph (use desmos or wolframalpha or whatever else you prefer), we can see that as we're getting closer to $0$, the distances between peaks and valleys of the cosine get arbitrarily small. But this also means that no matter how small we choose $\delta$, there are a peak and a valley which are closer to each other than $\delta$, so within the interval containing both, $f$ will never vary by less than $2$ (height of the peak to depth of the valley).
The other way around is also not true: uniformly continuous functions need not be bounded. Consider $f:\mathbb R\to\mathbb R,~f(x)=x$ as a simple counterexample.
However, what is true, and is also a pretty damn important thing to keep in mind, is that if $f$ is defined on a compact set and is continuous, then it is also uniformly continuous. This result is used in proving the fundamental theorem of calculus, for instance. But also more advanced stuff like Cauchy's generalized integral formula in complex analysis make use of this result.