I've got one of each. First let me say that I have heard variation between individual examples of soapdishes and string-thrus that were almost as different as the difference between the two. Individual guitars vary a lot.
Soapdish Gibson Style - a nice mid rangey tone, but with enough presence not to sound muffled. Some examples have a lot more sustain than others. It is possible to find these very cheaply, as they appeared on Silvertones that are sometimes mistaken for cheapies.
String-Thru Valco Style - The David Lindley/Jackson Brown "Running on Empty" tone when overdriven. Great for screaming distorted rock guitar sound. Dan Tyack loves these for sacred steel. My latest one also has a surprising amount of high end compared to the last two I played. (Note: the tone control came wired counter clockwise on some of them, so check it when you listen to the tone.) The sound is a little less delicate than the soapdish, but higher output.
You can't lose with either. They're both great, just a little different. That's why I have both.
-Paul
ymmv
(edited for speling)[This message was edited by Paul Arntson on 13 October 2005 at 11:17 AM.]