Are DFAs with a unary alphabet strictly less powerful than DFAs with a binary alphabet?
Take the set $A=\{2^k \mid k \ge 1\}$. The language 
$$ L=\{ \text{bin}(a) \mid a \in A\} = \{10^k\mid k \ge 0 \}$$
is clearly a regular language, whereas 
$$L=\{ 1^a \mid a \in A\} $$
is not regular. Proving the latter is a standard textbook application of the regular pumping lemma.