CIELAB Color Clustering
Generally speaking, a change of color space is essentially a nonlinear transform that stretches and distorts the coordinates. The transform is also continuous, except maybe on a singularity line or half-plane. It does not preserve the distances, but preserves the neighborhoods.

In the case of RGB > Lab, you can see it as a linear transform (change of basis, from RGB to XYZ) followed by a nonlinear transform of every coordinate independently (rescaling of XYZ to Lab). The topology is essentially preserved.

On the other hand, image segmentation algorithms do their best to delimit clusters of similar colors and separate the clusters from other clusters. So if you just deform the clusters, a good segmentation algorithm will adjust the boundaries and somehow undo the transform. If you repeat the experiment with XYZ or YUV coordinates, you should see no big difference either.

A stronger qualitative difference can be observed if the topology of the space is modified, for example by moving to the HSV space (Cartesian to cylindrical coordinates), of by performing some projection to lower the dimensionality such as XYZ > xyz. But beware of the physical interpretation of what you are doing.