For instance, the absolute value function is defined and continuous on the whole real line, but its derivative behaves like a step function with a jump-discontinuity.
For some nice functions, though, such as $e^x$ or $\sin(x)$, the derivatives of course are no "worse" than the original function.
Can I say something that is typical of the derivative? Is it typically not as nice as the original function?
Yes, that is completely right. And inversely, integration makes functions nicer.
One way of measuring how "nice" a function is, it by how many derivatives it has. We say a function $f\in C^k$ if it is $k$ times continuously differentiable. The more times differentiable it is, the nicer a function is. It is "smoother". So if a function is $k$ times differentiable then its derivative is $k-1$ times differentiable. A function is "as nice" as its derivative if and only if its smooth (infinitely differentiable). These are functions like $\sin(x), e^x$, polynomials, etc.
Inversely, integration makes things nicer. For example integrating even a non continuous function results in a continuous function: Is an integral always continuous?
The concept you're talking about is called smoothness (Wikipedia, MathWorld).
Functions like $e^x$ and $\sin(x)$ and polynomials are called "smooth" because their derivatives, and $n$th derivatives are all continuous. Smooth functions have derivatives all the way down, so they're as nice as their derivatives.
But functions like $\operatorname{abs}(x)$ and $\operatorname{sgn}(x)$ aren't smooth since there are discontinuities in either them or their derivatives. They're nicer than their derivatives.
A function is in class $C^k$ if its derivatives up to, and including, $k$ are continuous. So the number of levels of nice-ness will depend on $k$. Think about how integrating $\operatorname{sgn}(x)$ gives you $\operatorname{abs}(x)$, which in turn gives $\operatorname{sgn}(x)x^2$, and so on. As Zachary Selk points out, you can make functions nicer by integrating them.
In fact, for most functions, its more likely that they can be integrated than differentiated. Not only is being "nice" a rare trait, being differentiable at all is too.
You can view it as a consequence of convolution being at least as nice as the involved functions. Because an integral of a function $f$ is exactly that, a convolution :
$$\int_{-\infty}^t f(\tau)d\tau = \int_{-\infty}^\infty f(\tau) H(t-\tau)d\tau = (f*H)(t)$$
Where $H(t)$ is the Heaviside step function:
$$H(t) = \cases{0 &, t < 0 \\ 0.5 &, t = 0 \\1 &, t > 0}$$
As others have pointed out, derivative operators generally decrease the smoothness of a function. This is true for many classes of function spaces: Sobolev, Besov, Triebel-Lizorkin, etc. In fact most of these spaces have a smoothness parameter that decreases by applying differential operators. Terrence Tao has a nice diagram illustrating the relationship between these spaces on his blog.
One thing that I did not see mentioned in the other answers is that derivatives can improve the decay of a function. So if you start from a smooth function that grows, its derivative could be considered as nicer than the original.
So I would argue that differentiation creates a trade-off between smoothness and decay. This is most clearly seen in the Fourier domain, where differentiation is basically multiplication by a polynomial. Suppose the original function is $f$ and its Fourier transform is $\widehat{f}$. In the Fourier domain, a derivative of $f$ is given by $P(\omega)\widehat{f}(\omega)$, where $P$ is some polynomial. If $\widehat{f}$ has a singularity, it can be killed by the zeros of the polynomial, thus causing the derivative of $f$ to have a reduced rate of growth. At the same time, $P$ grows at infinity, which means the derivative of $f$ will be less smooth.
