Let's assume we have two trading systems, and for the same period of time they gave us two different sets of trades:
First system:
+1, +1, +1, +1, +1, +1, +1, +1, +1, -8
(nine +1 and one -8, resulting +1)
Second system:
-1, -1, -1, -1, -1, -1, -1, -1, +9
(eight -1 and one +9, resulting +1)
In terms of a returns distributions, the first system has a strong negative skew (like an option selling strategy), the second one has a strong positive skew (like an option buying strategy or a trend-following strategy).
Which system of these two is better?
If we use Sharpe ratio to compare, measures will be alike. If we use Sortino ratio, the second system will be an absolute champion because the strong positive trade will be ignored in volatility measure, only small negative trades will be counted.
But let's take a look at what really matters: an overall profit and a maximum drawdown. Results are the same for both of systems: the profit is +1 and the maximum drawdown is -8.
The difference is the first system drawdown is made by one strong loss, the second system drawdown is made by a serie of small losses. But does it really matter for you? Mathematically, it doesn't. Psychologically, maybe it does, if you want for some reason that you drawdowns grow steadily (to give you more time to enjoy it).
One possible advantage of the Sortino ratio coming to my mind is when your risk-management is so bad that a strong loss from a negatively skewed trading system is able to give you a critical damage. The Sortino ratio gives a bonus to positively skewed systems and penalizes negatively skewed, so it does for you some preliminary sorting of strategies ranking them by the danger they have for your bad risk-management. But here you shift the problem from one area (risk management) to another (trading system quality assessment), which I think is a bad idea.
Personally I continue to use the Sharpe ratio despite an abundance of various different measures.
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