one for the road
When Dean and his brother Sam first came into Anna's life she didn't really gravitate towards one more than the other. Both were there to equally give her what she needed: help and understanding. Help because the armies of Heaven and Hell were out to get her, and understanding because the world thought she had lost her mind. (When Dean acknowledged upon first meeting her and hearing her story that she actually was hearing angels and not crazy, her "thank you" was so honest it remains one of my favourite Anna moments.)
The balance between Anna and the Winchesters changed when Dean finally got a chance to talk with Anna alone after she regained her memories of her past life. He was having a hard time living with all the horrible things he did in Hell and found himself questioning life and emotions and the whole shebang, and he finally had a chance to sit down and have an honest conversation with an angel (something he could never have done with Castiel or Uriel and their brilliant people skills). Anna didn't have all the answers, but what did happen was they each found someone they could open up to and relate to - always a welcome thing in such hard times.
Later it was Anna who approached Dean, and although there was more deep and meaningful conversation, it was clear that wasn't all Anna was after when she surprised him with a kiss then stole his best line - pulling the "last night on earth" card. Dean could never resist that, so to put it as gracefully as Uriel did, he cut himself a slice of angel food cake.
But in the end the connection they had wasn't enough to come between Dean and his loyalty to his brother. Although co-operating with Uriel was part of Sam's plan to bring the angels and demons together and have them fight it out, there was still a very good chance things could go wrong and Anna would end up dead. The fact that he was hitting the bottle so early in the day was probably because Dean was aware of this, so when he agreed to hand Anna over to the angels in exchange for Sam's life I'm sure a part of him felt like that was really what he was doing. Hell, until Ruby showed up miraculously with Alastair at the last minute that's what he had done.
Anna forgave Dean even before the plan succeeded, but her return to angel-status seemed to herald the end of their brief romance. Although when Anna finally reappears it's to try and talk Castiel out of using Dean as a weapon of torture, her concern for Dean is mixed in with the whole issue of right and wrong and trying to get Castiel to side with her. In fact, Anna doesn't even try to talk to Dean at that point, and the only time they actually do see each other before the apocalypse kicks in Anna is strictly business and stops Dean in his tracks at the first sign of a come on.
So are they meant to be?
Moreover, the time Dean spent with Anna started a change in him that could be seen most clearly through Dean's reaction when he realized that the angels were actually fighting for the apocalypse. Instead of choosing to create a paradise on Earth where there was no pain or guilt or sadness, he chose to fight for his right to keep the bad with the good because he came to understand that's what being human means. This was a complete turn-about from his earlier desire to be rid of feelings that he had first expressed to Anna and later to Sam.