Using a lambda expression versus a private method

I read an answer to a question on Stack Overflow that contained the following suggested code:

Action<Exception> logAndEat = ex => 
{  
    // Log Error and eat it
};

try
{
    // Call to a WebService
}
catch (SoapException ex)
{
    logAndEat(ex);
}
catch (HttpException ex)
{
    logAndEat(ex);
}
catch (WebException ex)
{
    logAndEat(ex);
}

My question is: what is the advantage (if any) of using a lambda expression for LogAndEat as opposed to the (in my view simpler and more obvious) private method as follows:

private void LogAndEat(Exception ex)
{
    // Log Error and eat it
}

Edit: Thanks for the answers so far but just restating my fundamental question a little more clearly: which approach is better/would you recommend in this instance? A lambda expression or a private method?