I was surprised at how tough it was to track down anything that clearly mentioned both.
Firstly, from several recipes, including this one, it seems that simple syrup is made, as you might expect, by dissolving sugar into an equal amount of water and then cooling.
When it comes to invert sugar, recipes seem to call for more sugar and possibly an acid, be it cream of tartar or acidic acid (sometimes via lemon juice) to help catalyze the inversion of the sugar. On top of that, they often call for boiling until a temperature of 236 °F (114 °C). This process effectively breaks downs the bond between the glucose and fructose molecules that make up the sucrose.
This page explains that water activity (correlated with spoilage) is better minimized by monosaccharides, while simple syrup generally only contains sucrose, a disaccharide. It then goes on to describe a method for making "partially inverted sugar syrup" and calls either for acid or more sugar, as well as boiling for 20 minutes. Since this process is less exact and does less to force the system to break down the sucrose it's easy to see why it's "partially" inverting the sugar.
So to directly answer your questions, these are different things with simple syrup effectively just being sugar (sucrose) dissolved in water and invert sugar/syrup being dissolved sugar that has been processed to break down into it's component sugars (glucose and fructose). Invert sugar should be slightly sweeter, thicker, and have a longer shelf-life.