Assuming there is a normal distribution and not assuming so

Hi, I have a problem which I don't really understand.

I have 2 sets of data, which have been gotten from 2 different methods of testing something.

The data is in this case:
1 2 3 4 5 6 7 8
Method A: 212 180 197 204 219 258 206 200
Method B: 225 187 195 209 225 260 207 210

So, the ideal case would have been that the values for 1 were the same for both methods, the values for 2 were the same, etc.

Now, what I need to do is test for a difference between the 2 methods, both with and without assumption of a normal distribution, and as it turns out, I am quite clueless. If anyone could do some sort of walk-through on how to solve this, or atleast point me in the right direction, I would be most grateful.
If you are concerned with the difference between means between these two methods, you can use a paired t test assuming a normal distribution. When you assume that it is not a normal distribution, you can use a Wilcoxon signed ranks test.

You can find whether the distribution is normal or non gaussian by running a normality test in spss or any other advanced statistical package.

Hope this helps.