I read an answer to a question on Stack Overflow that contained the following suggested code:
Action logAndEat = ex =>
{
// Log Error
I guess you could think of the lambda expression in this example as being a little bit like Pascal's nested functions, in that it's code that can only be executed by the method it's declared in.
A private method could be called from any method in the same class, while a lambda like this is local to the current method and is therefore explicitly only used in that context.
That's really the only advantage I can think of - being explicit in terms of expected usage.