According to one book tearing paper towels on Shabbat is not allowed. However, tearing a tea bag to open it for use is permitted as long as one does not destroy any letters and as long as the tearing is done in a destructive way so no mending would be possible. Why doesn't the same reasoning imply that one should be able to tear paper towel as long one does not tear along the perforation?
You're on the right track, but there are some differences between paper towels and tea bags. Tea is food. You are allowed to tear open a package that contains food, so long as the action you do is not constructive (that is, you are not creating a new vessel by opening a sealed package). In the event that you are creating something new in the process of accessing the food, this action would be prohibited.
With paper towels, there is no such benefit. Destroying in order to create something (a pre-set paper towel) is prohibited biblically as an act of building (or a derivative thereof). Destroying for destruction's sake is prohibited by rabbinic injunction.
Along the lines you suggested, however, toilet paper may be torn on Shabbath (for its intended purpose) so long as one does not tear it on the perforation. In this way, one is not "building", and the rabbinic prohibition against destroying is waived for the sake of human dignity (ie., not to be forced to spend the day either in pain or covered in filth). But, again, no such leniency exists for (standard) paper towel use.
The difference between the tea bag and the towel is that when you tear the tea bag you want the food inside the bag, not the bag itself. In the case of paper towels you want the paper towel from the roll. This is called a melacha. The other case isn't because the tearing is only to get the food.
The principle is discussed in the gemarrah, chagigah 10a,b. If you dig a hole for the dirt, but not to create a hole, this is not a biblical prohibition. Either you could say it's purely destructive or is a melacha she'aino tzricha le'gufo which are both permitted bibically. In the case of the tea bag, you can rip it because you want the tea, not the ripped bag. The towel you want to use the towel, so it's tzericha le'gufo.
This is assuming of course that the paper towel is being used for constructive purpose. There is ground to stand on and say that if the towel is being used only to get dirty and throw away, that this isn't really constructive purpose but rather it's just to clean something up and would not be melacha. So I'm only answering what seems to be the book's reasoning, not to give you over what you should do in your case. Of course for psak in your case you should ask someone in person to hear the whole case, can't paskin from a book but that appears to be the book's logic IMHO.