付钱is to say you pay for the goods or servece you get,alway means that you give the money to sb. 交钱is to say you give your money to a school or institution for the education or judgement or sth like that you get(such as tuition,legal fea).

付钱 means pay money for the commodities or services you buy while 交钱 means kind of submitting to some organizations such as schools for tuition, government departments for fees of water, gas etc.

付钱and 交钱 there are little difference in our daliy life.but if distinguish them as you wish that 付钱 means that you can get the goods or servece at once when you pay them and 交钱 means that you should pay first or pay after the servece such as the water ,electricity, gas, education，fee of mobil phone and so on.

They have the similar meaning. But let me understand, 付钱 is in more formal way and sounds better.