The difference is what distribution the number of swaps has in both of the methods.
uniform cross-over:
we pick a gene to be swapped with a probability p independent from the other swaps, i.e. a Bernoulli experiment and we perform this Bernoulli experiment, for the whole chromosome, i.e. let's say for n genes, so the number of swaps will follow a Binomial distribution.shuffle-cross-over:
we randomly shuffle the chromosome first(this is done mainly to avoid positional bias, i.e. to decouple the probability swaps from the position of the genes in the chromosome -- we do take care of this bias in the uniform case too. What's different from the uniform case, is that we pick only one cross-over point, and all elements on one side of this cross-over point will be swapped, so we swap any number of genes with an equal probability of 1/2. Therefore we avoid also the so called distributional bias, i.e. probability of number of swaps being different.