Tables are a designers mainstay and offer a lot of control, once you get the hang of them. The problem with going beyond the limits of tables and the simpler align and style HTML tags lies in your target audiences? web browsing software and its ability to correctly interpret your designs.
Layers fall into one of those slightly questionable areas where browser software does not always render them the way you intended them to work which can mangle your page elements.
This is also the main reason why the majority of designers have not completely moved over to CSS page formatting to make faster loading pages with far more control over all page elements than by using more traditional methods. In theory at least, CSS and/or layers should give you, the designer, ultimate control over the final appearance of your site and by seperating content from presentation you get faster page loads and far more control over everything on said page. In practice though, the very real problems some mainstream web browsers have in correctly rendering CSS and layers means your pages can sometimes end up by looking like a sack of soot tied in the middle instead of the slick design you had originally intended.
I agree that using layers allows a lot of control and the fact that you can also add them (and their contents) to timelines for even more control only gives the argument in their favour more weight.
Try creating a heavily layered and/or CSS backed site then look at it with NetScape 4.7
If you don't blow a capillary I'll be very surprised.
Unless you can absolutely guarantee your target audiences browser software (an intranet over LAN is a good example) you have to design for the majority which is not necessarily the easiest method or the one that offers you the most control over your design.