In one of the recent comments section of Godol Hador's blog, he is bewildered over the concept of the "weak atheist" and does not understand the qualitative difference that it has from the term "agnostic."
Weak atheism is the idea that one does not know if there isn't a god but that one doesn't reject the idea either. The weak atheist is simply unpersuaded by theistic arguments. The weak atheist can also go by the term "agnostic atheist" which means basically the same thing.
This idea is different from simple agnosticism in that an agnostic is just making a statement of knowledge, not belief. An agnostic, while saying that they don't know (or cannot know), may also believe in gods or not.
In common use, the term agnostic has taken the "middle ground" between belief in gods vs rejection of that belief. But, in truth, no such middle ground exists. Look at the word "atheism." 'A' means 'without,' 'theism' means 'belief in an intervening deity or deities.'
One is either of a belief in theism or without it. You can waver back and forth between the two, but you cannot both hold a belief and not hold that belief at the same time.
Suppose someone offers you ice cream, you can either accept the offer, reject it, or stop and think about it. If you accept it then you have ice cream (i.e. theism). If you reject it then you are without ice cream (i.e. atheism). If you stop to think about whether you want ice cream or not, you are still without ice cream (i.e. _still_ atheism). The guy isn't going to give you your scoops unless you say yes. One cannot somehow have ice cream and not have it at the same time. Either you accepted the offer, or you did not. Not accepting is not the same as rejecting, but it still puts you in the same category of those without ice cream.
The atheist doesn't need to explicitly reject the idea of god, the atheist is still an atheist if he just doesn't accept the idea.