One of the things to look at is the interest rate you would be paying for either loan and how that would effect the total cost of the loan. Also, there is the question of the utility of any money spent up front. For example, if using a loan on the existing house would result in no up front costs and a 5% interest rate over 30 years, and the standard mortgage would cost $20,000 and have an interest rate of 8%, you're almost certainly better to use the existing house as backing and throw that same $20K in a long term interest bearing investment (e.g. government bonds). All this assuming you plan to hold onto the second property long term.
Compounding interest is a fantastic tool and a fearful master. If you can make it work for you, then do it. If you are facing the possibility of paying it, you almost always want to lower it as much as possible.