The Sharpe ratio has a real advantage over alpha. Remember that standard deviation measures the volatility of a fund's return in absolute terms, not relative to an index. So whereas a fund's R-squared must be high for alpha to be meaningful, Sharpe ratios are meaningful all the time.
Moreover, it's easier to compare funds of all types using the standard-deviation-based Sharpe ratio than with beta-based alpha. Unlike beta—which is usually calculated using different benchmarks for stock and bond funds—standard deviation is calculated the exact same way for any type of fund, be it stock or bond. We can therefore use the Sharpe ratio to compare the risk-adjusted returns of stock funds with those of bond funds.
As with alpha, the main drawback of the Sharpe ratio is that it is expressed as a raw number. Of course, the higher the Sharpe ratio the better. But given no other information, you can't tell whether a Sharpe ratio of 1.5 is good or bad. Only when you compare one fund's Sharpe ratio with that of another fund (or group of funds) do you get a feel for its risk-adjusted return relative to other funds.