Let's break these down, starting with the atheist claim:
There is no evidence for God, therefore No God.
1) To say that there is "no" evidence for God is simply inaccurate. In both the legal and scientific domains, evidence evaluated based on it strength, which is ultimately subjective (unlike mathematical/logical "proofs"). It is far more accurate to say that one is not convinced by the evidence presented.
2) concluding the non-existence of something based on the lack of evidence for it is the
argument from ignorance . One can only make an
epistemic claim from the lack of evidence—
that they don't believe that God exists .
Now the theist claim:
There is no evidence against God, therefore God.
1) We have the same problem with the claim of "no" evidence against God being an inaccurate claim.
2) Same argument from ignorance problem. But when it comes to belief...
3) There is an additional problem with this one. If we treated this like a scientific inquiry, then we would need evidence BEFORE we accepted the conclusion; we don't accept the conclusion (that God exists) simply because we have not found evidence against it. The evidence must support the conclusion for it to be accepted provisionally. However, in practice, we accept conclusions all the time without evidence, rather we simply trust the source. This is generally a good heuristic (depending on the sources), but when it comes to any important issue (like our eternity), we should ditch the heuristic and be more methodical in our approach.
Based on #3, I would say that the theistic statement is more fallacious.