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      One definition of the Omega ratio is:
 (Expected Return above threshold)/(Expected return below threshold), or
  E[ MAX(0, R-tau)]/E[ MAX(0, tau-R)] 
 where
    R = return( a random variable) of the portfolio,
    tau = a specified target or threshold return 
 Now, 
  MAX(0, R-tau) = R-tau+MAX(0,tau-R)
 so in terms of expectations
     E[MAX(0,R-tau)]/E[MAX(0,tau-R)] 
  = (E[R] -tau + E[MAX(0,tau-R)])/ E[MAX(0,tau-R)] 
   = (E[R] -tau )/ E[MAX(0,tau-R)] + 1
   Notice the similarity to the Sharpe ratio:
      (E[R] -tau )/SD( R),   where SD is Standard Deviation. 
So the Omega ratio and the Sharpe ratio effectively differ only in how risk is measured in the denominator.
Effectively, the Sortino ratio uses the semi-vriance as the risk measure.