For all intensive purposes, there is no difference between the terms podiatry and chiropody. They are the same thing. Some like to invent differences between them to make out that podiatrists are better than chiropodists for egoistical reasons.
Chiropody is the older name for the profession of podiatry. ‘Chiro-‘ means ‘hand’ and ‘pod’ means foot – in the past at the origins of the profession, chiropodists did treat hands and feet, but as the profession evolved, it was feet only. Though some do argue that the term could have meant that chiropodists were ‘handy’ with the feet, so the term was still appropriate.
The USA changed the name to podiatry a very long time ago. New Zealand and Australia changed later and the United Kingdom are the most recent country to formally change the name, though the terms chiropody and chiropodists are still used frequently in the UK. You almost never hear those words in the USA, Australia and New Zealand.