The answer is that once either effects activate, it takes the inline css property "display=none" off of the element. These hide effects require the display property to be set to "none". So, just rearrange the order of methods and add a css-modifier in the chain between fade and slide.
$('#anotherDiv').hover(function() {
$('#myDiv').stop(true, true).fadeIn({ duration: slideDuration, queue: false }).css('display', 'none').slideDown(slideDuration);
}, function() {
$('#myDiv').stop(true, true).fadeOut({ duration: slideDuration, queue: false }).slideUp(slideDuration);
});
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