Surface plots vs. colour maps
As everything, it depends. Here is a 3d plot of a mass-spectroscopic peak in time and mass to charge:


You can get a pretty good idea of its shape and how important noise is. Also, as it is a wireframe, you can see what is behind. If you want to see several of them I can show you too:



Now the thickness is not so obvious, but still, you get a nice idea of the whole thing. You can immediately see the relative intensities, and patterns like the uppermost corner, where there are several peaks parallel to each other (which is an important feature of my data), with falling intensities.

For the show off, I generated a bigger slice with Blender, that gives a nice feeling of how the data looks like, including the long "ridges", parallel peaks, noise, and profiles:



In this case, I don't care some peaks are covering each other because the exact positioning is quite random. If I do a heatmap I get this:



I can overlay more information, like the red lines on top, but now we have lost track of the relative intensities, actual shapes, and level of noise.

In my case, the 3D works because the position of the peaks is quite random, and the exact shape varies from experiment to experiment. Also, intensities and noise are important, and one needs to keep them in mind. But if you want to plot a function (say, a kernel density), where the exact shape is important, and relative values are not so much, a heatmap or a contour plot are usually a better choice.