I have degrees in both :). An undergrad in CE and graduate in IE (specialization in Human Factors Engineering).
Hard as it may seem to believe, depending on what IE specialization you choose, IE could come in handy. A friend’s kid is doing a PhD in CE and was told due to the nature of his research that he should pick up an MS in IE along the way (he’s doing research in optimization of waterways or some such obscure area). CE is very broad, as others noted, but invariably it focuses on building big, expensive stuff. IE, on the other hand, applies to all kinds of things (I use my skills for user interaction / user interface development).
If you like to make and break things and enjoy labs, the outdoors, and the like, CE will be great. If you spend hours thinking how to improve the workflow at Wendy’s, IE will be great. I’ve done both, and these days prefer the subtleties of IE work versus the hard hat CE work.