What are Search Indexes?
A search index contains a list of products along with meta-data describing the products. Amazon created search indexes to speed up searching (that's obvious from the name). It appears that each search index can have different meta-data based on the type of product and that Amazon could develop special search algorithms specifically for each search index. But they seem to have done this only with the Books index which is why Power Search only works on the Books index.
A browse node may be associated with lists of products. But instead of relating a browse node to products, Amazon associates browse nodes with product entries in search indexes. So, to search for products associated with a browse node, you have to specify a search index. Browse nodes may have associated products in more than one search index.
The Product Group returned by A2S is used by Amazon to determing how to display the product's information. It's used to determine what information about the product to display based on the type of product.