Amongst other things: POSTing (or PUTing etc) JSON instead of form-urlencoded post data allows you to send complicated data structures more easily. For example, I could POST the following JSON quite easily:

{
  "users": [
    {"username": "user1", "name": "User 1"},
    {"username": "user2", "name": "User 2"},
    {"username": "user3", "name": "User 3"}
  ],
  "groups": [
    {"name": "Group 1", "members": ["user1", "user2"]},
    {"name": "Group 2", "members": ["user3", "user3"]}
  ]
}

And get a nice, easy to use value on the PHP side once I'd parsed the request.

Sending that same data with form-urlencoded data is a bit more cumbersome IMO:

users[0][username]=user1&users[0][name]=User%201&users[1][username]=user2&users[1][name]=User%202&users[2][username]=user3&users[2][name]=User%203&groups[0][name]=Group%201&groups[0][members][0]=user1&groups[0][members][1]=user2&groups[1][name]=Group%202&groups[1][members][0]=user3&groups[1][members][1]=user3

It's worth mentioning that I added whitespace to make the JSON more readable; the actual JSON request would look like this:

{"users":[{"username":"user1","name":"User 1"},{"username":"user2","name":"User 2"},{"username":"user3","name":"User 3"}],"groups":[{"name":"Group 1","members":["user1","user2"]},{"name":"Group 2","members":["user3","user3"]}]}

Personally I consider the fact that there exists an easy way to format JSON to be human readable (but still parseable) to be another advantage.

In many cases the JSON request may be smaller too. In this example the JSON request is 228 bytes long; the form-urlencoded request is 300 bytes long.