It wasn't until Hawkeye met and fell in love with Mockingbird, a very unthinly veiled imitation of Black Canary, that Hawkeye began to evolve. No longer just the team's pointman, Hawkeye stepped up and became leader of the West Coast Avengers, a role he would take up off and on until Mockingbird's(and the series') death. Later, when working with the Avengers under another's (even Cap's) leadership no longer suited him, Hawkeye went to the Thunderbolts, helping the reformed villains gain the trust of the public and leading the team into becoming a real fighting force. He even gave up his freedom when he was arrested for aiding and abetting fugitives.
When he finally returned to the scene, he was quickly killed in the worst death scene written in Avengers Disassembled, after making a point that the Avengers never did enough to make sure an enemy or situation was harmless. Basically, they would take the win and go home, instead of making sure whatever they were dealing with wouldn't come back and bite them in the ass. While dead, his name was taken up by the Young Avenger Kate Bishop , who was given the name by Captain America himself, saying the mantle needed to be taken up by someone worthy.
House of M brought the archer back to life, but in a time while everything he knew was shot to hell. While currently not an archer, Hawkeye is still in the fight as Ronin in New Avengers. But, come on. How long will he resist the siren call of the longbow and how long before too many people ask how the hell did he get so good with a sword?
So who wins?
As far as series goes, Green Arrow has an established history of maintaining a series for a respectable period, but I don't think Hawkeye has ever been given a real shot with top notch talent, like Green Arrow has in his recent series (Meltzer may be cringe worthy now, but Archer's Quest was brilliant).
Unlike Green Arrow who has been stuck in the same role since the seventies, Hawkeye has shown real versatility as a character without losing any believability in his development. Hawkeye can be the hot-head who has fistfights with USAgent, the Joe Everybody leader who commands loyalty through trust, the smart alec showoff cracking jokes or the guy who questions without being a hypocrite. He's like your older brother, best friend and wiseass friend all rolled into one.