What is the relationship between magnetic field quantities $B$ and $H$ in cgs units? In cgs units permeability of free space is dimensionless. If so, then in cgs units won't $B$ and $H$  have same units as they are related by equation:
$B=\mu_o H$
But instead why they have different units such as Gauss and Oersted?
 A: The same reason why in SI reactive power measures in volt-amperes, but not in watts: for convenience and because of historical (=occasional) reasons. It is convenient when you can say what quantity — $B$ or $H$ — someone means when they say "XXX gausses" or "YYY oersteds". But there is nothing tricky here, and $1\,\text{Gauss} = 1\,\text{Oersted}$.
You can think about this unit names as two aliases for one unit with a set of additional informal rules in what situation which name should be used. Although some purist might frown if you say they are same unit, that's the worst consequence.
A: There are multiple cgs systems, so my comments are specifically about Gaussian units.
It is correct that in Gaussian units $B$ and $H$ have the same dimension, however nothing prevents a unit system from having multiple units with the same dimension. For example, in SI units $\mathrm{Hz}$ and $\mathrm{s^{-1}}$ have the same dimension but the former is used for frequencies and the latter for time constants. Or, $\mathrm{rad}$ and $\mathrm{\%}$ have the same dimension but the former is used for angles and the latter is used for parts of a whole.
Similarly, in Gaussian units $\mathrm{G}$ and $\mathrm{Oe}$ have the same dimensions but the former is used for B fields and the latter for H fields.
Note that although they have the same dimensions it is useful to keep them separated. In a linear isotropic medium, the formula that relates them is $$\vec B =\mu \vec H$$ where notice that it is $\mu$ the magnetic permeability of the linear isotropic material and not $\mu_0$ the vacuum permeability. Indeed, $\mu_0$ does not exist in Gaussian units, and $\mu$ is dimensionless. So these Gaussian units are not generally equal even though they have the same dimensions.
