So I actually e-mailed customer service and asked this question a few weeks back. Let's pick on the Prodigal Son, since that is the hilt I was asking about. Basically, they told me if you want to go with DIY electronics, ASP is the way to go. They did not give me a reason. I bring up the Prodigal Son since you can basically build or order the same hilt on both sites.
Also, you will probably (and I emphasize *probably*) get your parts faster with ASP. I usually end up waiting the normal 6-8 week or longer lead time for my hilt if I order from the site.
The ASP hilts tend to be more cookie-cutter whereas the hilts on the site can have special requests placed in the notes. So for example, I recently asked if a Covertech knob not be drilled into an ASP hilt, but was told they are pre-drilled. If I order from the site, I can request this and it will come without the Covertech Wheel drilled.
I hate to say, "it depends", but it depends on how fast you want, what you want, and if you want any customizations. Also, it is worth noting that not all requests can be honored in an order from the site...but they tend to be more flexible. Customer service is great about working with you on this.
I hope this helps.