Reconstruction is trying to literally follow the way the old Celtics did by looking at historical information and not the more modern fluff. The Victorian historians romanticized the Celtic and their religion. Reconstruction tries to fix that.
Neo-Druidry is a modern version of what the Celtics believed. It is changed up to fit with today's society and tries to pull from some sources that reconstruction does...but it can still contain a lot of information that is incorrect.
There is also Celtic Wicca. This is the Wiccan faith mixed with Celtic concepts like the gods, ogham writing and divination, tree calendar, and etc. It is in no way traditional to what was practiced at the time of the Celtics.
Nothing we have today will ever be exactly what the Celtics believed. A lot of the info was destroyed when the Romans and Christianity came through. Most of what we have is from monks writing down the legends and from what we have rediscovered from Northern France sites.
Therefore, most people I know that follow a Celtic tradition just go with a general Celtic spirituality. Meaning they take Celtic practices and beliefs and add it to their own personal spirituality as they see fit.