A responsive website is a website that resizes automagically based on the size of the device being used. The website "knows" to show the full site on your desktop, a smaller version on your tablet, and a thin version on your iPhone/Android. It has nothing to do with the content of the site or the speed of your site.
Mobile optimized means that the assets for the website are designed to be used easily on mobile. For example, if you have a big image on the home page that's 50kb (a fair size for a home page image), it would be fine in terms of loading time and bandwidth for your desktop computer. However, 50kb might be a lot for an iPhone to download and it might take longer for the iPhone to download that size of a file. So "mobile optimization" here would be to either a) show the iPhone a 10kb file instead of a 50kb file, or b) replace the image with nice text on the iPhone instead.
So in summary, responsive design describes a website that "shrinks to fit" on different devices. Whereas mobile optimized refers to the concept to minimizing the download time and bandwidth that would be needed to browse your website on a mobile device (the idea being that for mobile, we want smaller file sizes because mobile devices have slower connections/processors/RAM, etc. vs desktop computers).