In the original, the Charmed Ones gain their powers through the reading of the Dominus Tritus spell. In the remake, the sisters' powers are unbound by their mother, Marisol Vera. The Vera/Vaughn Sisters each have a different father; while in the original, the original trio all shared the same father.
In the original, whitelighters weren't meant to reveal themselves to their charges, unless it was necessary. In the remake, it appears whitelighters are allowed to reveal themselves to their charges when they meet.
In the original, whitelighters teleported by orbs. In the remake, they have a different teleportation power.
whitelighters in the original series didn't have telekinesis while the whitelighters in the remake do, or at least Harry does.
In the original series, the Halliwells weren't given a choice of whether they become witches. The remake gives the assumption the Vera sisters do have a choice and have 48 hours to choose whether they take up their powers
There is an Easter Egg in the Vera Book of Shadows that stems back to Melinda Warren, the founder of the Warren/Halliwell witch bloodline from the original series. Currently, it's unknown as to whether Melinda in this new series is an ancestor to the Vera family.
Jenna's name might be derived from the short-lived character of Jenny Gordon. Jenny was the niece of Piper's third love interest in the original series, Dan Gordon. Jenny was written out of the original show because of negative feedback from fans who deemed the character unnecessary and annoying.
Mel and Maggie went back in time after casting a spell to find out what type of evil spirit they are dealing with by revisiting the spirit's past. In the original show, time travel was a recurring theme.