You've got the definitions right. The difference is simply that between absolute convergence and conditional convergence.
If all terms in the sum are nonnegative, then there is no difference between the values of these two sums: either both of them are the same finite number or both are $\infty$.
If $\sum_{n\in\mathbb{N}} |a_n|<\infty$ then $ \lim_{N\to\infty}\sum_{n=1}^N a_n = \sup\left\{\sum_{n\in I} a_n\,:\,I\text{ finite} \right\} $ provided all terms are $\ge 0$.
If some terms are positive and some are negative and the sum of the absolute values is finite, then $\sum_{n\in\mathbb{N}} a_n$ is defined by treating the positive and negative terms separately, and then adding those two sums.
It is only when the sum of the positive terms and the sum of the negative terms are both infinite that there can be a difference. In that case the limit of finite sums may be a finite number, but it then becomes a different finite number if the order in which the terms are added is altered. The most well known example is the alternating harmonic series $ \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} =\log_e 2. $ If you rearrange the terms, you can make the limit of finite sums a different number. For example, if you add enough positive terms, the sum will exceed $10$, then add one negative term and the sum is less than $10$, then add more positive terms until the sum is more than $10$, then the next negative term, etc., and then the limit of partial sums is $10$.
Later note: Another example is that $ \int_0^\infty \frac{\sin x}{x}\,dx $ exists as an improper integral, i.e. a limit as the upper bound of integration approaches $\infty$, but does not exist as a Lebesgue integral because the integrals of the positive and negative parts both diverge to $\infty$.
An easier example is $ \lim_{a\to\infty} \int_{-a}^a \frac{x\,dx}{1+x^2}\text{ versus }\lim_{a\to\infty} \int_{-a}^{2a} \frac{x\,dx}{1+x^2}. $ The values of these two expressions can be found by freshman calculus methods, and they're different from each other.
Another, somewhat more involved, example that yields to freshman calculus methods is $ \int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \, dx \, dy \text{ versus } \int_0^1 \int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \, dy \, dx. $ Work them out and get two different numbers: $\pi/2$ and $-\pi/2$ (I don't recall which is which). Conclusion: The integrals of the positive and negative parts of this function must both diverge to infinity. This is an example of the difference between interated integrals and double integrals. These are iterated integrals. The double integral is an integral with respect to Lebesgue measure in the plane; the iterated integral is an iteration---one integral inside another---of two or more integrals with respect to Lebesgue measure on the line.