The Electro Acoustic
Put simply, this is an acoustic guitar with a pickup placed under the bridge or around the sound hole. The idea of an electro acoustic is to keep or even enhance the acoustic tonality of the guitar being played. The pickup enables you to plug the guitar into an amplifier (it will sound even better if the amp is designed for an acoustic guitar), thus giving you the ability to 'taylor' the sound to your requirements. Other benefits of an electro acoustic are that you can add effects such as reverbs and delays to really enrich the sound. Some players will find the acoustic more enspiring by plugging it in and adding effects, others will want to keep the sound as 'original' as possible.
The Semi Acoustic
This guitar is not an acoustic at all. Think of it as an electric guitar with slight acoustic qualities. The acoustic quality comes from the fact that the body, which would normally be solid on an electric, is cut out, usually either side of the pickups. One of the original semi acoustics was the Gibson ES335. The trademark being the 'f' holes either side of the pickups. These 'f' holes are a cosmetic feature but the holes inside give the guitar a more 'boxy' acoustic resonance. This gives the semi acoustic a slightly different sound than the standard electric guitar. Some semi acoustic guitars are not immediately obvious as the sound holes can be covered up. I own a model called a Fender Flame (this guitar inspired Robben Ford who now has his own Signature model with the same 'acoustic chambers'). To look at, you would think they were a standard electric but when you plug in, you may (!) notice the difference