The theists trust their own experience or the recorded experiences of others to believe God exists.
The atheists trust their own experience or the recorded experiences of others to believe God doesn't exist.
Atheists use logical argumentation to support their contentions. Interestingly, they adduce evidence that has questionable value with respect to their arguments. I'm sure you'll have no problem finding lots of analyses showing that atheists' arguments are logically unsound in terms of both the validity of the premises and the logical leaps that atheists make to get to their conclusion. I've even presented a few here and there on this forum. You'll find accessible analyses along these lines in book reviews of some of the more popular recent restatements of the old atheist arguments, some of which have actually been best sellers of late.
One of the oldest (as in hundreds of years old) formal logical arguments against the existence of G-d is known as the "
Argument from Evil." There are four variations of it:
1)the argument from imperfection
2) the argument from natural evil
3) the argument from moral evil
4) and the argument from unbelief
Each of the four arguments from evil begins with the claim that if God existed then the world would reach a certain standard. The standard anticipated differs between the different forms of the argument, each argument claiming that the evil named in its title—imperfection, natural evil, moral evil and unbelief respectively—would not exist in a world created and sustained by God.
http://www.philosophyofreligion.info/?page_id=52
Religionists also have logical arguments that presumably prove the existence. The main difference between the atheists and religionists is that atheists are totally dependent on logical argument because they haven't had an experience of G-d or any other experience that leads them to believe.
You can't very well argue from nonexperience because it does not necessarily reflect on the potential object of the nonexperience (e.g., G-d). That is, you can't say G-d does not exist just because you've had no experience of Him or because you choose to believe others who report not experiencing G-d.
Talk about not seeing the forest for the trees. Everyone bickers about the details, and yet the spectre of common human cognition and behavior looms before us all...
I think the religionist and atheists are miles apart in terms of world view, ethical framework, and cognitive style. These people are basically living in totally different worlds.