Primarily tone, aesthetic, and depth. Of course they didn't copy the systems 1:1, but it _feels_ more like diablo than diablo 3 ever did. I realize that I'm using vague terminology, but you'd be hard pressed to find fans that don't agree.
We're talking about a "spiritual successor" here, not "add up how many aspects of PoE are identical to D2." I know you said D1, but few people make or care about that comparison. D2 is the most popular and the latest entry in the series that people who were turned off by D3 care about.
I'd describe PoE as more of a spiritual successor to Diablo 2. There's a lot of weird aspects of D1 which were removed in D2 and later games -- for instance, spellbooks, elixirs, and randomly generated quests.
As a concrete example: PoE's schema for items (normal/magic/rare/unique, maximum of 3 prefixes and 3 suffixes) is almost a carbon copy of Diablo 2's item system.
We're talking about a "spiritual successor" here, not "add up how many aspects of PoE are identical to D2." I know you said D1, but few people make or care about that comparison. D2 is the most popular and the latest entry in the series that people who were turned off by D3 care about.