For both statements, we will be using a very simple dictionary: Gx - x is a god.
First, the statement, "There is a god."
(∃x)Gx
Translated, this reads, "There exists an 'x' such that 'x' is a god." In order for this statement to be true, one needs to find only one example of something that is a god. In order for this statement to be false one would need to examine every single object in the domain and ascertain that it is NOT a god. As the assumed domain in this case is 'everything', that is a helluva lot to look at.
Proving this statement to be false is therefore impossible, whereas proving this statement to be true is (in theory) simple.
Now lets look at the next statement, "There is no god."
~(∃x)Gx
Translated this reads, "It is not the case that there exists an 'x' such that 'x' is a god." For this statement to be true, one would have to examine every object in the domain to ascertain that it is not a god. For this statement to be false, however, one simply needs to find a single example of an object in the domain that is a god.
Proving this statement to be true is therefore impossible. However, proving this statement false is, again, simple.