The words are similar. In AmE, I would suggest that which is attractive can be appealing and visa versa. There is a slight differentiation between the two, however, in that "attractive" implies some visual aspect to the "attractiveness". The visual aspect may not account entirely for why you think something is attractive but for something to be attractive some aspect of it should be visually appealing. Things that are attractive have a tendency to "turn your head" so you can take a look at them.
Appealing, on the other hand, suggests that the appealing object affects you emotionally, spiritually, visually, etc. Things that are appealing have a tendency to "speak to you" or "draw you in".
So it would be odd to say, "Does the orchestra sound attractive?" Better to say, "Does the orchestra sound appealing?'
One might say, "Do you find the chrome on my motorcycle attractive?" One might also say, "Do you find the chrome on my motorcycle appealing?" If one finds the chrome attractive they overtly mean it's pleasant to see. If one finds the chrome appealing they might add chrome to their motorcycle or desire a ride on yours.