Just finished watching this...
I had seen the odd episode of this show before, but I just finished watching the 'essential' episodes based on one of the various lists that exist, and may eventually watch the rest. These are my thoughts based on what I saw:
1. Anakin - I guess this is the obvious aspect that drew most people to this show. Anakin is a better character as seen through the eyes of this show. I especially loved the hint we get in the Mortis episodes that he was for a while very skeptical and possibly even bitter about the whole 'chosen one' thing. I suspect he didn't really believe in it or want it, despite his general confidence and cockiness, which isn't something we ever really learn in the films - actually we hear pretty much nothing from him on the whole issue. We also see how he still feels anger about having been a slave, another thing the films ignored. And finally, most importantly perhaps, we see his gradual disillusionment with the Jedi Order. When he confesses to Ahsoka that he's thought about leaving, that moment gave me chills.
2. Jedi and (romantic) relationships - I'm not the world's biggest romance fan in general, but this was a very interesting aspect to me. We actually get to see it play out through three different Jedi. (1) Obi Wan and Satine - this isn't a hugely explored relationship, but the most interesting part to me was the end when Maul attempts to provoke Obi Wan to anger and revenge...and fails. Obi Wan remains firm in his convictions, saying "It takes strength to resist the dark side. Only the weak embrace it." And he truly lives by those words. I had more respect for his character than ever after this, because we learn he's been through just as much as Anakin and yet never turns. And we all know what would have happened to Anakin had he been in Obi's place. (2) Anakin and Padme - obviously this was more covered in the movies than in this show, but I like it's portrayal in the show. They are loyal to each other in spite of their relationship being tested by circumstances, which isn't something we see in the films until it comes time for their relationship to end. (3) Ahsoka and Lux Bonteri - Apparently a lot of people don't like this pairing. I'm in the minority who do. It makes complete sense to me because Lux is a novelty to Ahsoka, who has spent her life surrounded by warriors and soldiers. It's a cool role reversal. And I thought it was interesting the way it was handled. She chooses, in the end, to let him go. She's a teenager and yet she shows more maturity that Anakin when it comes to staying true to her Jedi vows.
3. Villains - this was one of the best parts of the series. I enjoyed how morally grey some of the bad guys are, and how even the ones who are all bad have depth. Maul is a great example. He barely speaks in TPM yet in the show he becomes a truly compelling character with his own struggles. Ventress was another favorite of mine. She's driven by anger and hatred like any dark side user and yet she has moments of compassion. I loved the introduction of the night sisters and all the hidden agendas. We never really are sure if they're good or bad. And I loved Ahsoka's final arc with Bariss as the villain. She's a great character and a great example of how those with good intentions can fall far.
4. World building - another GREAT aspect. I loved seeing the depth in the world that this series offered. We see neutral worlds, separatists who are good people with differing views, the seedy underbelly of the capital and the way ordinary people live, the sheer number of different agendas of people in power...it's fantastic.
I may come back and add more thoughts here when I have time. Overall, great series.