Leaf shape, primarily.
In the end it is very nuanced. The faces of two kind-of similar-looking people are easy to differentiate. It is less easy to explain where the faces differ.
In itself, kurume hybrids differ, as they are hybrids. There's a spectrum, and the spectrum overlaps. There's hybrids of R.indicum and R.kiusianum and in that case the kurume vs satsuki becomes blurred.
It is also flower shape, leaf texture, stamen, growth habit, colour patterns. The bark of R.kiusianum is rough. R.indicum and most satsuki are more smooth.
In the end, you need pictures or you need to go look at plants themselves.
Looking at the pure species is often useful, as all hybrids are different, as they are hybrids.
Maybe this post is helpful:
http://www.bonsainut.com/threads/j-t-lovett-azaleas.22574/#post-350998
Remember that species are homozygous for most alleles. That means they have the same gene twice for most traits, and that trait is expressed. When you cross two different species, all genes are heterozygous, two alleles for each trait. So two different genes for each trait. The dominant trait will express. This is called F1.
When you cross again, you get F2. And for each trait, there is 25% odds for a recessive trait, 75% for a dominant trait. So you can get all combinations of all traits.
And genetics isn't truly Mendelian, so it's not just one gene, one trait, and dominant vs recessive. But thinking about it in simple terms is often useful.