Well, a lot actually! More than you’d probably think.
At the core, Celtic Recon is based in historical and archaeological research authentic to the particular Celtic culture dealing with. (There are several Celtic Cultures. For example, I am a Gaelic Recon, which means I study the Gaelic culture which is today modern day Ireland, Scotland, and Isle of Man.)
Meanwhile, neoDruidry is a post-Enlightenment Period interpretation of Celtic cultures heavily based in romanticism themes and faulty information. This isn’t to say that neoDruidry has no meaning, though.
NeoDruidry is, from what I’ve gathered when I researched it, a mostly (if not completely) different religion from what the Celtic people practiced. Some neoDruid organizations incorporate the Greco-Roman elements (Earth, Fire, Wind, Water) in their practice while those elements are not found in Celtic archaeology (at least not Gaelic.) The more research-based organizations like ADF aren’t Celtic but Indo-European, so calling that organization Celtic would be misleading.
This isn’t to say that neoDruids don’t incorporate Recon-like research to better understand their practices or “roots”, but just that they don’t place that much importance on it when creating their own practices (if they did, they wouldn’t call themselves druids in my opinion.)
Here’s some helpful links:
Also, there are several large neoDruid organizations and each are a bit different in their approach towards neoDruidry. Despite deciding that neoDruidry isn’t for me, I really enjoyed John Michael Greer’s book The Druidry Handbook. It gives a brief explanation towards the history of the “Druid Revival.”