What tests can I use to effectively test if the distribution of values within two matrices significantly differ? How can I test if the distributions of values within two matrices significantly differ?
I want the weight of the values to be considered when testing, so that two matrices uniformly filled with 1's are not significantly different but a matrix uniformly filled with 1's is significantly different to a matrix filled with 1000's.
Example from my data of two matrices I want to test the difference between:


I am assuming you have multiple instances of each matrix, and you want to test whether the distributions are different. Otherwise, what you need is just a distance between them, like the Frobenius norm of the difference, or any other norm which suits the problem you are working on.
If I understand correctly, you also don't want a multinomial test (e.g. Pearson's $\chi^2$) because of your requirement that a matrix of all 1s and one of all 100s should be considered as originating from different distributions.
A simple non-parametric test is the permutation test (this is directly out of Wasserman's book "All of statistics", Chapter 10). Let's say that you have $n$ matrix samples $X^i \sim F_X$ and $Y^i \sim F_Y$. You want to test
$$ H_0: F_X = F_Y \ \ \ \  \text{vs} \ \ \ \ H_1: F_X \ne F_Y.$$
Let $T(X^1,...,X^n,Y^1,...Y^n)=|\bar{X}-\bar{Y}|$ (the norm of the difference of the means of the first $n$ matrices minus the mean of the second $n$ matrices). Compute a value $T_j$ for each of the $(2n)!$ permutations of the inputs and let $t$ be the value of $T$ observed in the original data. Then
$$ p\text{-value} = \mathbb{P}(T > t)=\frac{1}{(2n)!} \sum_{j=1}^{(2n)!}I(T_j > t), $$
where $I$ is the indicator function. Because computing $(2n)!$ values is very costly, a MonteCarlo approximation can be used (i.e. compute just a fraction of the permutations). You can also take any other statistic of the whole data, as long as large values can be used to reject (or you will have to reverse the condition in the $p$-value).
