You asked why they are separate and the answer simply is that's what the law provides. Most custody orders do NOT require approval for moving. If this one does, a contempt can be filed. Note that a good lawyer would help a lot here. I recently got (and it wasn't hard) child support enforcement to dismiss their case because support was being addressed in another case, and since support can be addressed in a modification, a good attorney will help you a great deal.
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You need to check your custody order. In most cases, the spouse does not need approval. However, in family law, divorce may be permanent but custody and child support are not necessarily so. Though they are separate actions, a parent may move for modification of support or custody based on a substantial change in circumstances. Please contact me if you need any help.
Divorce Child support Child custody Divorce and family Child support order Child support and custody Child support modification Child support and changes in circumstances Child support enforcement Enforcing child support by contempt Enforcing interstate child support payments Lawsuits and disputes Family law Contempt of court