Sorry I can't be of more assistance here. Did a little digging around the JED and it APPEARS that there may be a plugin required for the default registration with Recaptcha that was discontinued with Joomla 1.5 as it wasn't native.
I'm using Recaptcha myself but with the Jomsocail registration and it's been great. Have a look if noone else chimes in and be sure to let us know how you do.
Just tried another template and the recaptcha works fine so somehting in the solarsentinel template is causing the conflict.
This is not the first conflict I have had with this template and unless anyone can figure out how to overcome this, I guess I will have to try another template. Might be the best option in the long run.
I'll check out the extension you suggested anyway,
Sorry again, not to be of more help, but I can tell you that I'm using RECaptcha with this template with no issues. It'd be a shame to change templates because of this as your site looks great and I'm sure there's a lot of work in there.
There are other captcha's in the JED and personally, I'd try at least a few before thinking about switching templates. I've used Big O Captcha for contact forms before and found it simple and straightforward, just not with the registrations.
Good luck and keep plugging! (that's not to intend a top-water plug... I'm a fisherman too)