Sample locations with a uniform pattern will create aliasing in the output, whenever there are geometric features of size comparable to or smaller than the sampling grid. That's the reason why "jaggies" exist: because images are made of a uniform square pixel grid, and when you render (for example) an angled line without antialiasing, it crosses rows/columns of pixels at regular intervals, creating a regular pattern of stairstep artifacts in the resulting image.
Supersampling at a finer uniform grid will improve things, but the image will still have similar artifacts—just less badly. You can see this with MSAA, as in this comparison image from an NVIDIA presentation on temporal antialiasing:
The 8x MSAA image (which is not quite a grid but still has a repeating pattern) still plainly has jaggies in it, although they're antialiased jaggies. Compare to the TXAA result, which both has a higher effective sample count (due to temporal reuse) and uses a Gaussian rather than box filter for accumulating the samples.
On the other hand, random sampling produces noise instead of aliasing. There is no pattern to the sample locations, so no pattern to the resulting errors. Both aliasing and noise are errors due to not having enough samples to form a clean image, but arguably noise is the less visually-objectionable artifact.
On the other other hand, perfectly random sampling (in the sense of i.i.d. random variables) tends to exhibit a certain level of clumping. Purely by chance, some areas in the domain will have denser-than-average clumps of samples, and other areas will be lacking in samples; those areas will be, respectively, over-represented and under-represented in the resulting estimate.
The convergence rate of the Monte Carlo process can often by improved by using things like stratified sampling, low-discrepancy sequences, or blue noise. These are all strategies to generate "de-clumped" samples that are spaced out a bit more evenly than i.i.d. samples, but without creating any regular patterns that might lead to aliasing.