The difference is the 380m is based on the older Arrandale core. The i3-2310 is slower clocked but based on the newer Sandy Bridge core. They both probably perform similiar and consume the same power but the Sandy Bridge based cpu will have better integrated graphics.
The obvious difference is the clock speed. The Core i3 380m is clocked at 2.53GHz. The Core i3 2310m is clocked at 2.1GHz and is based on the newer Sandy Bridge core; clock for clock it is about 10% faster than the previous generation Core i3. Therefore, the Core i3 2310m is equivalent to a Core i3 380m clocked at 2.31GHz. Technically, it will be a minor decrease in performance.
The real difference is the graphics core. The Core i3 380m relies on an graphics core/card that must be integrated into the motherboard. In a budget laptop that would mean you will get an Intel GMA 4500MHD. It is not really meant for games, but people have used it to play games at low settings with somewhat decent performance I suppose. For better gaming experience you would need to buy a laptop with an AMD or nVidia graphics core.
The Sandy Bridge Core series comes with a new graphics core that is integrated into the CPU itself. The Core i3-2310m has the Intel HD 3000 graphics core. It is at least 2x more powerful than the Intel GMA 4500MHD. It is about the equivalent to the desktop version of the Radeon HD 5450. It is capable of playing Crysis with low settings and no anti aliasing.