We recently purchased a home and would like to know which item will keep our home cooler/warmer and reduce our bill – solar screens or tinted windows?

It depends on which screen and which tine you are comparing.  The critical factor is how effective they are.  The effectiveness is measured by either the Shading Coefficient or the Solar Heat Gain Coefficient.  In both cases the lower the number, the less heat is allowed to transfer through the window.  Generally speaking, shade screens have a Shading Coefficient of about 0.35 and most tints have a Shading Coefficient of between 0.5 and 0.75.  So in general, shade screen are better than most tints, but there are always exceptions to the rule and there and many new exciting tint products on the market that a great at keeping the heat out.  So it is better to contact a solar screen company and a widow tint company and check out all products before you make a selection.