The problem with Conjugate[] is that it is a Numerical Function:

In[80]:= Attributes[Conjugate]

Out[80]= {Listable, NumericFunction, Protected, ReadProtected}

and it doesn't seem to work on symbols. The same is true of Re[], Im[] and Abs[].:
E.g..

In[2]:= Head /@ {a, b}

Out[2]= {Symbol, Symbol}

In[1]:= Re[a + b I]

Out[1]= -Im[b] + Re[a]

That doesn't even make sense. Why would the real part of a complex number include the negative of the imaginary part of symbol b. Apparently, trying to take the real part confuses MMA so that it can't compute correctly. Hence,

In[11]:= z = a + b I

Conjugate@z

Out[11]= a + I b

Out[12]= Conjugate[a] - I Conjugate[b]

If Conjugate[] knew that a and b were Reals, it would know that Conjugate [Real x] =x, etc. So I devised my own simple function:

In[77]:= z = a + b I

conjugate[z_] := z /. I -> -I;
conjugate[z]

Out[77]= a + I b
Out[79]= a - I b

..useful for more complicated expressions:

In[61]:= a E^(I x) a conjugate[E^(I x)]

Out[61]= a^2

The Assumptions should not even be necessary here; FullSimplify[Conjugate[a] Conjugate[b] - Conjugate[a b]] by itself will work.

ComplexExpand[Conjugate[a] Conjugate[b] - Conjugate[a b]]