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N0. It is NOT the case that it is true in general. It is true for all cases with both $m \le 5$ and all prime $p n \lt 100$but may always be false . It is also true for $m=6$ and $n = 31$. But actually it is false ( which requires for $p\ge 31.$)

At m=6$) over$\mathbb{Z}$and also over$\mathbb{Z}/{n}\mathbb{Z}$for any rate$n \ge 36$, a specific example is the prime or composite. The sets $$A=\{0, 1, 4, 10, 12, 17 \} \text{ and } B=\{ 0, 1, 8, 11, 13, 17\}$$ for$n= 37.$Here are Sidon sets with$A-A=B-B$is everything in$A \subset \mathbb{Z}/{37}\mathbb{Z}$except \mathbb{Z}$. So the only restriction on $\{14, 15, 18, 19n$ is that they are Sidon sets. Since the elements of $A-A=B-B$ range from $-17$ to $17$, they are Sidon sets with $A-A=B-B$ in $\mathbb{Z}/{n}\mathbb{Z}$ for any modulus, 22prime or not, 23\}.$starting with$n=35.$It can does not work for$n=32,33,34$. It does also work for$n=31.$However for$n=31$we have$B=11A.$Also, for$n=35$we have$B=17A-11.$These are the only cases where an affine map takes$A$to$B$. I found this example looking in$\mathbb{Z}/{n}\mathbb{Z}$but it turns out to be checked that known. I think it could be decided if there are no any exceptions for$m,b$with m=4,5$ (other then the special case $B=\{ma+b \mid m=4$ and $n=13$) however the method seems tedious. Here is a \in A \}.$There sketch: We may be assume that$A=\{0,1,a,b\}.$If$B-B=A-A$then$s-r=1$for some construction here but I did not find one yet$r,s\in B.$We can translate to have$r=0,s=1$so that$B=\{0,1,w,x\}.$Further more,$w,x$are elements of$A-A.$Three of the 13 elements ($-1,0,1$) are ruled out, leaving$\binom{10}{2}=55$possible cases. Many of these (like$w=-x$,$w=1-x$and$w=x-1$) are immediately seen to be impossible. The ratios between elements rest come in pairs$A$\{0,1,w,x\}$ and $\{0,1,1-w,1-x\}$ which are all squares reflections of each other.

If we try $B=\{0,1,1-a,a-b\}$ then $A-A$ and $B-B$ overlap in $9$ elements leaving $\{b,-b,1-b,b-1\}$ unmatched in $A-A$ and $\{1-2a+b,-1+2a-b,a-b-1,1-a+b\}$ in $B-B$ If $b=1-2a+b$ then $n=2a-1$ and $B$ is seen to be a translated reflection of A. We can't have $b=1-a+b$ because $a \subset ne 1.$ The other two possibilities for $b$ also fail.

But if we try $B=\{0,1,-a,b-1\}$ there are $6$ unmatched expressions in each of $A-A$ and $B-B$ and running through the possible matchups of pairs one case is $b=-a-1,a-1=1-b-a$ which comes out to $a=3,b=-4$ which leaves the remaining elements as $\pm2,\pm 4,\pm 7$ for $A-A$ and $\pm2, \mathbb{Z}/{37}\mathbb{Z}$ but pm4, \pm6$for$B-B$. So this sub-sub-case is possible for, but only partially the case for,$B.$n=13$ where, it can be checked, $B$ is a translated diolation of $A$.

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N0. It is NOT the case. It is true for $m \le 5$ and all $p \lt 100$ but may always be false for $m=6$ (which requires $p\ge 31.$)

At any rate, a specific example is the sets $$A=\{0, 1, 4, 10, 12, 17 \} \text{ and } B=\{ 0, 1, 8, 11, 13, 17\}$$ for $n= 37.$ Here $A-A=B-B$ is everything in $A \subset \mathbb{Z}/{37}\mathbb{Z}$ except $\{14, 15, 18, 19, 22, 23\}.$

It can be checked that there are no $m,b$ with $B=\{ma+b \mid a \in A \}.$

There may be some construction here but I did not find one yet. The ratios between elements in $A$ are all squares in $A \subset \mathbb{Z}/{37}\mathbb{Z}$ but this is only partially the case for $B.$