This explanation is going to require a lot of maths. I will explain it all as simply as I can, but I apologise in advance for the sheer volume of it.
I think the main difference is that the sum of an infinite series doesn’t always exist. The easiest example is 1+1+… doesn’t mean anything. I suppose you could say that it was ‘infinite’, but it doesn’t converge (have a limit) in a traditional sense. Another example is 1 - 1 + 1 -1 + 1… As you sum terms you alternate between 1 and 0, so the sum cannot converge.
The problem is that, because you’re looking at infinite series, you have the misleading association with summing finite series. The issue of convergence should really be looked at from the perspective of sequences (or nets, but I’ll scare people if I start talking about nets. :)). A sequence is basically a way of assigning something to each counting number. e.g. 3, 3.1, 3.14, 3.141,… is a sequence with first term 3, second term 3.1, etc. We say this sequence converges to Pi. Convergence has a precise technical definition, which basically amounts to ‘a sequence converges to x if the nth term gets arbitrarily close to x as n becomes large’, but it must come close for all sufficiently large terms. For example, if I was to take that sequence converging to Pi and replace the 100th, 200th, etc. terms with 1 then it would no longer converge to Pi.
There are of course many sequences that don’t converge, such as 0, 1, 0, 1…
For the familiar numbers, which mathematicians call the Real numbers, the most important test to see if a sequence converges is that if there is a number such that every term in the sequence is less than it (we say that number bounds the sequence above) and each term in the sequence is greater than or equal to the last, the sequence converges to some number. This is called the fundamental theorem of analysis. As an example we have the sequence 3, 3.1, 3.14, 3.141,… Every term is less than 4, and each term is greater than the previous. Thus it converges to some number, which we call Pi.
Ok, now that’s out of the way I can answer your question. 
Say we have an infinite sequence x_1, x_2,…,x_n,… (I hope this notation is clear - it’s basically just the same as using variables x,y, etc. to represent numbers. We just have an infinite number of these variables so we assign a number to each). We define a sequence x_1, x_1 + x_2, x_1 + x_2 +x_3, … This sequence is called the sequence of partial sums. Any one of these terms behaves exactly like normal addition (because it is normal addition). We then define the infinite sum x_1+x_2+x_3+… to be the limit of this sequence.
It would be nice if infinite sums behaved exactly like normal finite sums and, as Ultrafilter said, in a lot of cases they do. Given a number x, we define the absolute value of x to be x ignoring the sign. i.e. if x is positive then the absolute value of x is the same as x, if x is negative then the absolute value of x is -x (which is positive, because x is negative). We denote this as |x|. An infinite sum x_1 + x_2 + … is called absolutely convergent if the infinite sum defined by |x_1| + |x_2| + … converges. Absolutely convergent sums are nice because they converge and you can rearrange the terms however you want and they will still converge to the same number (this isn’t entirely trivial to prove, and I’m certainly not going to do it here. Take my word for it). The bulk of infinite sums people use are absolutely convergent, simply because they’re so much nicer than the alternative and do in general behave almost exactly like a normal sum.
The easiest example of a sum which does converge but is not absolutely convergent is 1 - 1/2 + 1/3 - 1/4 + 1/5 - … If you rearrange the terms as 1 + 1/3 - 1/2 + 1/4 + 1/5 - 1/3 + … then it will still converge and it contains all the same terms as the previous, but it converges to a different number. The problem is that, while you can always rearrange each finite sum, you can’t rearrange all the terms at once. When you think about it, this isn’t really surprising - You shouldn’t expect that when you’re dealing with an infinite number of things it’s going to behave like a finite number. The problem is that there are so many nice examples that one grows to expect that all examples are nice.
Does that help?
Now that that hijack is over, the OP…
I’m more or less agnostic, but for the moment I’ll assume there is a god. If he is indeed omnipotent, I would say yes. Even if we still use the same axioms and rules of inference as now, he can consistently change our perception of these. As has been said, maths is a construction of humans rather than nature, and thus god cannot change it directly. However (we are for the moment assuming) humans are made by god, and thus subject to her influence. If she wished, 1+1 could be equal to 3 and we would all believe this and not percieve any trouble doing so, even if we also believed that everything was defined in the same way - ‘all’ that is needed is a consistently applied universal deception. Thus, if she wished, Pi could equal 3.1415926… or something equally silly instead of 4. 
