Obamas Shoes
Go for the natural gas. I think you can get more BTUs with the propane but I'm not sure. You have to worry about running out of the propane and that is no fun.

Mr. Grummp
Obama's Shoes is (are?) correct, you can pump more BTU's out of a tank of propane. Most grilling is done at lower temperatures however (Low & Slow), and with natural gas you don't have to carry around those heavy tanks. Oh, yeah, you asked a question! No, no difference in taste or food quality between the two.

David H
Mr Grump is a little reversed. Propane burns at a much higher rate than natural gas. You have the hook up already in your yard so use it to your advantage. Grilling is done at higher temperatures vs. bbq which is low and slow. The food doesnt taste different between the 2 gases but you will get a flame just a little higher with the LPG vs. natural gas because of the conversion but it wont be much of a difference. The choice is yours i use only charcoal and wood to grill and bbq. thats my preference but ive used both working at many restaurants throughout my cooking career and noticed very little difference. The big difference was between brands and styles.