Everyone is doing long lists explaining why they like one or the other, but nobody explains their choices. So here are my top 5 reasons why I think Potter is better than twilight:

1. Depth of character.

HP
Every major (and most minor) characters we meet in Potter have some deep background. This background explain current and past actions, thoughts, and beliefs - just as a background should.
We end up understanding Snape's feelings. We learn that his hatred towards James goes much deeper than the schoolyard. It goes as deep as love. We learn that he wants to take down voldemort, because after all this time, always.
Dumbledore also has a very deep background. We come to understand his obsession with taking down voldemort, and that it actually hits close to home.
Hermiones background is just as intricate. She comes from outside of the magical world, but in a different way than Harry did, giving her a different perspective and attitude.

Twi
We don't learn too much about the main characters. Sure we understand how the vampires came to be, but we don't learn what they do when Bella isn't around, or what they did before Bella. because of this we can't really justify their actions with their past.
Jacob is very one dimensional. What do we know about his past? Either then the fact that him and bella played together as kids.

2. The plot.

HP
There is a saying when it comes down to writing and it's this:
If you hang a gun on the wall in act 1, you must use it in act 3.
J.K Rowling does this countless times in the Harry Potter series. It adds depth to the plot, adds value to re-reading, and shows that the plot was well thought, and well constructed.
One example of this is the mentioning of a man named Nicolas Flamel on a chocolate frog card. When first read of this character he isn't at all important. But as we continue reading we find out more about him, and it is even pointed out to us that he was mentioned on the card (sending us flipping back to the where it is mentioned just to see if it really was).

Twi
The really aren't any guns on any walls in twilight. The only main one is Jacobs story of werewolfs and the cold ones. That is the only gun in the entire plot. We have no need to re-read in order to see the clues and allusions we missed the first time. The plot is thin, constantly leading from point A to point B, with no detours in between. Everything always works out and nobody ever dies.

3. The Message.

HP
Stories are told to entertain, and to teach lessons. The lessons can be hidden or they can jump up and hit you in the face, but either way, a good story teaches a good lesson.
Potter teaches us the importance of loyalty not just to others, but to yourself. When everyone else is calling you a liar, you need to be loyal to yourself. You know the truth and you need to fight for it. Don't sell yourself out just because it is the easy thing to do.
It teaches us to stand up for what is right. Sometimes, the other side has a better, powerful, and bigger team, but that doesn't mean you should back down. stand up and fight because it is right. Don't back down because it would be the safest and easiest thing to do.
It teaches self sacrifice for the greater good. There are times when you may need to sacrifice safety, comfort, or even yourself in order to help the greater good. That doesn't mean go and chuck yourself off of a building to make a stand for something. But that does mean you need to be brave and prepared to do extreme things for the greater good.

Twi
There aren't too many lessons in twilight. The main lesson it teaches (apart from how important it is to have a boyfriend) is standing together and backing your friends in the time of need. The cullens have helped many people, they have done good in the world, and as a result their friends come and stand behind them in their time of need.

4. The female leads.

HP
The importance of a female lead is to add diversity to the story, and to show people, especially young women, that they don't need men to help them be strong.
Hermione and Ginny are strong examples of this. They both fight for what is right, they don't take no for an answer, and they don't let the men become over bearing.
Hermione practically leads the hunt for horcruxes, and when Ron leaves she never gives up.

Twi
Bella is hardly what you would call a strong female lead. Whenever Edward isn't around she can hardly function. When he leaves her she shuts herself off from reality for 4 months, and then becomes reckless in an attempt to hear Edward's voice.
The one time she stood up for what was right despite what Edward said was when she wanted to keep her hybrid child.

5. Tragedy

HP
All good stories need tragedy. It's unfortunate, it sucks, it's...well, tragic. But it adds believability. When fights happen people die. Good people die, it's a fact of life.
One of the saddest moments in Harry Potter was when George Weasley died. He died during an epic battle of good vs. evil, and there isn't a more honourable way to die.
An angering death in Potter was that of Sirius Black. Sirius died because Harry jumped into danger without thinking. Sirius came to Harry's aid and died as a result. Sirius became another martyr in Harry's life, assisting in fueling the fire that was Harry's fight.

Twi
Nobody important dies in twilight. Sure James dies, but that's it.
During the newborn battle of eclipse, everyone walked away with their life. In a battle like that people will die, and not just the bad guys.
During the final confrontation in breaking dawn, there isn't even a fight let alone a death.
It doesn't mean meyer is more attached to her characters, it means she can't let go in order to add believability.