What is the difference between "differentiable" and "continuous"

Differentiability is a stronger condition than continuity. If $f$ is differentiable at $x=a$, then $f$ is continuous at $x=a$ as well. But the reverse need not hold.

Continuity of $f$ at $x=a$ requires only that $f(x)-f(a)$ converges to zero as $x\rightarrow a$.

For differentiability, that difference is required to converge even after being divided by $x-a$. In other words, $\dfrac{f(x)-f(a)}{x-a}$ must converge as $x\rightarrow a$.

Not that if that fraction does converge, the numerator necessarily converges to zero, implying continuity as I mentioned in the first paragraph.

Differentiability means that the function has a derivative at a point.

Continuity means that the limit from both sides of a value is equal to the function's value at that point.