The main difference between the original Battlestar Galactica series and the remake is in tone. The remake was much darker than the original, and much more "realistic" about how characters would react to the destruction of the 12 Colonies and the loss of billions of lives. In the original series the characters were kinda sad for about fourteen minutes and then, Wow! Look! A space casino! Not to say that the original series didn't have its darker moments (the Count Iblis storyline was pretty dark, in a fairy-tale way) but there was just no gravitas. It was a different type of show, a space opera adventure instead of a SF drama.
The other major differences are the religious and political was much more important in the new series – religion was a huge subtext, and the jockeying for power resulted in elections and an attempted coup. Religion obviously also had a big part in the controversial ending, involving Angels and a frequently criticized "God did it" aspect.
When you start to look at other variations: the different sexes for some characters, the "replicant" Cylons, the Final Five, the introduction of several new characters such as Laura Roslyn and Chief , the loss of original series characters such as Daggitt, Serina and the Imperous Leader… It's simpler to document what was DIDN'T change than what was different.
What was the same (between the two series):
- The 12 Colonies (and destruction of same)
- The original Cylon designs
- The original Viper designs
- The Battlestar Galactica and Pegasus surviving the attack
- The Adama family
- A hotshot pilot named Starbuck
- A ragtag fleet searching for Earth
- A character who betrays humanity named Gaius Baltar
- A fight for survival
- Supply rationing as a plot point
- A kid named Boxey (only in the miniseries)
In my opinion, the remake was the textbook example of EXACTLY what you want to do when you remake anything – you keep the core of the original, and add depth and your own take on the material. Of course, where the debate occurs is what you define as "core," which is why many people don't like JJ Abrams' Star Trek film (because it changed too many things that were "core" to those fans/critics).