I have a really tough time with this.
The problem I find is that I can only really judge Bond by one of two criteria, either by the villain he's up against, or the plot that unfurls before him. After all, a hero is only as good as the villain he's up against or the obstacle he has to overcome.
On the first count, in terms of opposition, from a statistical standpoint, I have to give Roger Moore the nod, and not just because one of them was played by Christopher Lee, or that another one was played by Julian Glover, or that another was played by Yaphet Koto, or that another was played by Christopher Walken... you get where I'm going with this. Connery's Bond has Dr. No, Goldfinger, and Blofeld, but the evil denizens of Moore's world were equally bombastic and diabolical, yet distinctly had more of them. So, I have to give the nod to Moore, with honorable mention going to Pierce Brosnan, who went up against my favorite villian, none other than Elliot Carver as played fantastically by Jonathan Pryce.
On the second count, if we go by the all-important story, the challenge to overcome, I'm inclined to give it to Connery with Thunderball, but he loses his place on the list because of that insufferable remake Connery practically force-fed us in pitiful admission that he never had a gig as good as Bond. Then again, he does get back on the list with my equally favorite 'classic' Bond film, You Only Live Twice. It's got rockets, it's got ninjas, it's got pihranas, it's got sumo wrestlers, it's got one of the most truly tragic deaths of all films (the accidental poisoning of his faux bride)... I could go on and on. Then again (pattern?) there's also Tomorrow Never Dies where the evil force that must be stopped is (to date, really) the unusual and utterly unprecedented choice of a media conglomerate. It's kind of genius, really.
In the end, the game is a one-on-one showdown on the ice between Brosnan for Tomorrow Never Dies and Connery for You Only Live Twice, with Thunderball sitting in the penalty box for punching Moore in the teeth in the first three seconds of the first period. And, if I ever get sick of watching the stalemate, I just go and visit Moore in the infirmary or go for a drive/swim in the Lotus because he may not cause my jaw to drop and eyes to boggle like the others, but he won't disappoint me, either.