With Assert.Throws() you can be really specific. You can say: an exception will be thrown at this method.

Whereas if you use ExpectedException: you do not have that level of granularity: the exception will be thrown somewhere during the test.

Furthermore, with Assert.Throws, you can test the exception that is returned itself.

   [Test]
    [Category("Slow")]
    public void IsValidLogFileName_nullFileName_ThrowsExcpetion()
    {
        // the exception we expect thrown from the IsValidFileName method
        var ex = Assert.Throws<ArgumentNullException>(() => a.IsValidLogFileName(""));

        // now we can test the exception itself
        Assert.That(ex.Message == "Blah");

    }

Have you been paying attention?

If so please answer: why would you use Assert.Throws over ExpectedException?