Why might a monotone increasing but nonlinear transformation of a utility function not represent the same preferences? According to a textbook, a monotone increasing but nonlinear transformation of a utility function might not represent the same preferences. Why is it so?

An example of such preference would be appreciated.
One reason I could think of is regarding convexity of the function (and hence risk aversion). For example, $u(x) = x$ is risk neutral, but $v(x) = \ln(u(x)) = \ln(x)$ is risk averse.

See also this.