Making a sorbet by cooking or by pureeing the ingredients?
There are a few things to consider here. Firstly, sugar dissolves in hot water better than in cold water. So, regardless of other considerations, it's worth making a simple syrup of the water and sugar.

What will change by heating is a few things. From a taste standpoint, depending on how you cook the fruit, the sorbet may taste "cooked". If that's the flavor you're looking for, cook away! However, "fresher" tasting sorbets are made by just pureeing (and straining). Cooking the fruit also stops oxidation (and discoloring) which happens with some fruits. On the other hand, the acidic lime juice will help prevent oxidation. Speaking of lime juice, if it's freshly-squeezed, a large component to its flavor are the volatile citrus oils. Cooking will likely cause these to evaporate, thereby losing some "citrusy" flavor (the acidity will remain, though).

From a texture standpoint it depends on a few things. Cooked fruit is usually softer; when pureed it will contribute towards a smoother texture. Additionally, many fruits contain pectin which is released when cooking. Pectin will contribute positively to the resulting texture of the sorbet. However, overcooking does degrade the pectin.

One way to get the best of both worlds is to gently poach the fruit in the simple syrup to soften it and release the pectin. (Alternatively, you can sous vide the fruit.) If you really want the "fresh" flavor (my preference) but want the benefit of pectin you can add a hydrocolloid thickener like xanthan gum or guar gum in a small amount (usually under 1% by weight).