Downside deviation uses only returns below a threshold, usually zero or the risk-free rate. For a positively skewed strategy the Sortino ratio is noticeably higher than the Sharpe, and for a negatively skewed one it can be lower.
Worked example: annual return 12%, total volatility 15%, so Sharpe is 0.80 at a zero risk-free rate. If downside deviation is 9%, the Sortino is 1.33. Nothing improved; a different denominator was chosen.
The honest caveat is sample. Using only the downside observations roughly halves the data behind the estimate, so the Sortino is noisier than the Sharpe and easier to flatter by a sample containing few bad periods. Report both.
Related: sharpe-ratio, skewness, calmar-ratio, drawdown