The difference is pretty simple to understand mathematically, See since the INITIAL exposed to risk was developed within the binomial framework, the lives who die have to be counted as exposed to risk until the end of the year of age (not the investigation), whereas the central exposed to risk is developed within the poisson framework, it only measures the total time under observation (the difference being if a life dies, the life is only counted as at risk until the moment of death). So to summarise
if we let ai be the latest of the time of entry into observation, the date of attaining age label x, or the beginning of the investigation, and let bi be the earliest of death, date of losing age label x or the end of the investigation, we see that the initial exposed to risk is,
Ex = Sum over all lives (1-ai) - Sum over LIVES WHO DO NOT DIE (1-bi)
Whereas the central exposed to risk is
ExC = Sum over all lives (bi-ai)
Hopefully this should make some sense, intuitivley the initial exposed to risk really has no meaning, can only really be understood interms of the above equation.
The relationship between the two by assuming that on average deaths occur at age x+1/2. See by making this assumption we know that the length of time we will have to "add on" in respect of each death to the CENTRAL exposed to risk will be 1/2 for each death, since it has to be counted as exposed to risk until the end of the year of age. So this is where the Ex = 1/2dx + ExC comes from.