JFBConnect can set both the Open Graph tags (which work for Facebook and Google) and the Twitter Card tags which work for Twitter.
If your website logo is being used, it means that either JFBConnect is setting that as the image or that no 'better' image was found to be used. To fix this, please make sure you've enabled the Open Graph - Content plugin, which will let JFBConnect scan your Joomla article contents to find a better image to use.
Additionally, on any page, you can add the following tag to make JFBConnect use a custom image for sharing to any of the networks:
{SCOpenGraph image=http://yoursite.com/link-to-best-image.jpg}
I hope that helps explain, but if not, or you still have questions, please let me know a page that's sharing the wrong image and we'll gladly investigate further.
Thanks,
Alex