We can find some attempts for generalizing the notion of entropy of discrete random variables to random variables with general distribution function.
A straightforward way is to employ Riemann sum of the distribution function. So we start with a discrete random variable and then by making the intervals small enough, the entropy function is obtained. Denote the quantized random variable by $X_\delta$ where $\delta$ is the size of the intervals. If the probability density function $f$ is integrable, we can see for small $\delta$ (Cover-Thomas p. 248):
$$
H(X_\delta)\approx h(X)-\log\delta.
$$
By choosing $\delta$ equal to $2^{-n}$, i.e. $n$ bit quantization, we get
$$
H(X_\delta)\approx h(X)+n
$$
which represents how many bits we need to describe $X$ with $n$ bit accuracy.
This shows somehow the relation between differential entropy and discrete entropy. Note that when $\delta\to 0$, $H(X_\delta)\to\infty$.
Another point is that the mutual information does not change using this method, namely if $\delta\to 0$:
$$
I(X_\delta;Y_\delta)=I(X;Y).
$$
The generalization is attributed to different people, among them mainly Kolmogorov and Rényi:
A. N. Kolmogorov. On the Shannon theory of information transmission in
the case of continuous signals. IRE Trans. Inf. Theory, IT-2:102–108, Sept.
1956.
J. BALATONI and A. RENYI, Remarks on entropy (in Hungarian with English and
Russian summaries), Publications of the Mathematical Institute of the Hungarian
Academy of Sciences, I (1956), pp. 9--40.
Renyi introduced the following random variable ($[]$ is the integer part)
$$
X_n=\frac1{n}[nX].
$$
Note that this is nothing but looking at the intervals $[\frac kn,\frac{k+1}n)$.
Suppose that $H([X])$ exists, which is denoted by $H_0(X)$ in the original paper. The lower dimension of $X$ is defined as following
$$
\underline d(X)=\liminf_{n\to\infty}\frac{H([X])}{\log n}
$$
and upper dimension of $X$ as:
$$
\overline d(X)=\limsup_{n\to\infty}\frac{H([X])}{\log n}.
$$
Now if $\overline d(X)=\underline d(X)$, we simply talk about the information dimension of $X$, $d(X)$ and we define the following:
$$
H_{d(X)}(X)=\lim_{n\to\infty} (H(X_n)-d(X)\log n).
$$
Renyi proved that if $X$ has an absolutely continuous distribution with the density function $f(X)$ and finite $H([X])$, then we can say:
$$
d(X)=1\\
H_1(X)=h(X).
$$
This is what we discussed above for $\delta=\frac 1n$:
$$
H(X_n)=h(X)+\log n
$$
Kolmogorov instead introduced the notion of $\epsilon-$entropy which is defined for random variables in abstract metric spaces which is more general.
To answer your question, We can keep the same intuition as the discrete case for differential entropy at least when we use it for finding mutual information or KL-divergence.
For the entropy itself, we have to alter our intuition a little bit. The entropy of discrete random variable means the minimum bits we need to compress the random variable. But for random variables with uncountable supports, we can always "compress" it with another uncountable set of same cardinality (any one-to-one and onto mapping does that). But different random variables with uncountable supports can have different differential entropy.
Best Answer
I think there are two separate things going on here. One is the issue of a maximum entropy distribution. The other is of whether or not distributions are invariant under different parameterizations. Regarding the second matter, I think your statement "if we had chosen a different parameterization, we should clearly arrive at the corresponding distribution" is probably not quite right (I say probably because I may be interpreting you wrong). Only particular distributions have this property and sometimes are not probability distributions. See http://en.wikipedia.org/wiki/Jeffreys_prior if this is what you're interested in.
ps I'd have preferred to leave this as a comment, but can't yet I guess.