Factor 1000 in mixing ratio formula
The factor of 1000 may be related to the expected units. The mass 'units' can be found in the derivation.
The full derivation of mixing ratio goes like this
$$w=\epsilon\frac{e}{P}$$
since, by the ideal gas law,
$$e=\rho_vR_vT$$
and
$$P_d=\rho_dR_dT$$
then the mixing ratio is expressed as
$$w=\epsilon\frac{\rho_vR_vT}{\rho_dR_dT}=\epsilon\frac{\rho_vR_v}{\rho_dR_d}$$
Since $\epsilon=\frac{R_d}{R_v}=0.622$ (see above hyperlink),
$$w=\frac{\rho_v}{\rho_d}$$
To return back to the question 'what is the difference between your two equations?' The units reported are $g/g$ and $g/kg$ for the first and second equations, respectively.