Why can't people talk while inhaling?
Via deep scientific analysis (i.e. trying it myself 5 seconds ago), I have determined that you can in fact speak while breathing in, it just sounds funny.

Think of the vocal chords as being like the body of a flute. As air passes by them, they vibrate and make sounds. Through careful modulation of their shape, specific sounds can be reproducibly made (this is equivalent to opening and closing the stops on the metaphorical flute). The way the pressure in your throat changes when you inspire (breathe in) is very different from the way in which the pressure changes when you expire (breathe out). These pressure differences are a large part of the reason why you sound different when you speak during inspiration as opposed to expiration, just like breathing into a flute differently would produce different sounds (although potentially the same notes). 

Other physiological, neurological, etc. effects probably also cause differences in the vocal cords during expiration/inspiration and thus also contribute somewhat to the differences in the sound.