TIFF is more widely supported. Many programs don't deal with PSD because the format is very complicated. TIFF on the other hand is like a "standard" image format along with JPEG and PNG.
Both TIFF and PSD can preserver layers information. Both of them can handle 16 and 32 bit image. However PSD can contain much more than that. Since it is the native format of Photoshop, it can have many photo-editing metadata like layer styles, layer folders, snapshots, custom channels, and even editing histories (though histories may make the file really big).
Personally I'll just save TIFF, because it's not likely that I'll have, say, a hundred layers, and a complex layer structure. But if I'm designing, I probably want to have as much metadata as I want, and also tons of layer styles, which leaves me no choice but PSD.
EDIT: according to @Conor Boyd and @ysap etc., TIFF can actually contain as much information as PSD does. So ya, go with TIFF. However, note that TIFF's being a general purpose image format does not mean that every image viewer can read all the information stored in any TIFF image. A "baseline" TIFF reader, for example, may only render the first layer in the image. This behavior is allowed by the TIFF standard.