The most obvious differences :
Professional players (Non-independent) are usually a part of the National team and have all their expenses, salary and everything else paid by the Association. And usually they have their winning cut by the NA for something like commission.
Independent players are those who are not under any National Association and more or less self-financing their expenses for training or participating in Tournaments etc., and they will get all their winning prizes.
Anyhow, although Independent players paying everything themselves, they will still need to register (ask for permission) to enter the tournaments via their National Association and it is up to the NA to say yes or no. So the main difference between professional (non independent) players and Independent players are whether they are self-financed (not a part of the national team) or they are "owned" by the national team (expenses paid but the association have the control over the players timetable etc).