My personal favourite for this type of task is LogicalExpand (and I can't believe that hasn't been mentioned in the existing 5 answers...)
In[1]:= A = Array[a, {2, 2}];
B = Array[b, {2, 2}];
LogicalExpand[A == B]
Out[3]= b[1, 1] == a[1, 1] && b[1, 2] == a[1, 2] &&
b[2, 1] == a[2, 1] && b[2, 2] == a[2, 2]
This will also work just as simply in the case of the higher-rank tensors that you're interested in.
If you want the output as a list, then you could do a Block hack like:
In[4]:= Block[{And = List}, LogicalExpand[A == B]]
Out[4]= {b[1, 1] == a[1, 1], b[1, 2] == a[1, 2],
b[2, 1] == a[2, 1], b[2, 2] == a[2, 2]}