Basically they are the same thing. Pushchair is the term commonly used in UK English, while buggy is more used in US English. However, because of international communication etc, American English terms have become very popular outside the US, and particularly in the UK, so you often find the word buggy being used. But both words refer to a forward facing vehicle in which the baby will be more or less sitting upright, though supported.
You get a similar confusion with pram, stroller and even the old fashioned term baby carriage. These all refer to the larger vehicle, in which the baby lies down, and is more often used for newborns.
However, all these terms get used to apply to both types of conveyance, so it is not surprising that a lot of people get confused. Even the people who make them tend to mix up the words - the main thing is to find what you want and then call it what you prefer!