Are DFAs with a unary alphabet strictly less powerful than DFAs with a binary alphabet?

Are DFAs with a unary alphabet strictly less powerful than DFAs with a binary alphabet? Is this even a meaningful question?

For example, if $ \Sigma = \{\texttt{0}, \texttt{1}\}$ , we can encode any larger alphabet using $ \Sigma$ , but if $ \Sigma = \{\texttt{0}\}$ , this can define a DFA (that say, recognizes $ L = \{ \texttt{0}^k \mid k > 0\}$ )… but such a DFA would never be able to recognize more “complex” regular expressions. For example, there is no way to encode $ \texttt{0011}$ using a unary alphabet that a DFA would recognize (we could use, say, Godel numbering, but that would require a more powerful machine that could “count”).

If DFAs with a unary alphabet less powerful than DFAs with a binary alphabet, is there a name for this language/grammar? I recognize this is kind of an odd question, since the DFA that recognizes $ L = \{ \texttt{0}^k \mid k > 0\}$ recognizes all unary languages… but technically there still are a countably infinite number of DFAs in this class ($ L = \{ \texttt{0}^1 \}$ , $ L=\{\texttt{0}^2\}$ , etc.)

Note I am of course assuming that for $ \Sigma = \{ \texttt{0} \}$ , that it does not contain the empty symbol $ \varepsilon$ .