Most vegetarians who stop eating meat for ethical reasons also take measures to avoid contributing to the suffering and death of animals (and even insects) in other capacities. For instance, a number of vegetarians refuse to wear leather and silk because they see it as an ethical violation of their respect for all living things.
Vegetarians who refuse to wear silk argue that the process involves unnecessary cruelty to moth larvae. Rather than allowing the moth to grow and leave the cocoon, silk manufacturers boil the larvae alive, causing them to suffer and writhe, in order to obtain longer strands of silk.
Leather, by contrast, does not directly contribute to the suffering of animals in most cases. In most cases, leather is made from the byproducts of animals that would be slaughtered for meat, rennet, and other animal products.
This is precisely why many vegetarians who have an ethical dilemma with meat have no problem wearing leather: because they do not see it as the primary reason for killing the animals, but instead a byproduct of the slaughter.
However, certain groups of vegans oppose wearing leather on the grounds that it indirectly contributes to the suffering of animals.
These vegans argue that contributing money to the companies that own the slaughterhouses (and sell the byproducts for leather, etc.) is just as bad as actually purchasing and eating meat yourself because you are still contributing money to the continuation of institutionalized animal suffering.
This is certainly something to consider if you are currently a vegan or a vegetarian for ethical reasons. It may have been tough to give up meat in the first place, but if you are truly committed to the cause and you believe the arguments are strong-enough, you may want to avoid clothing purchases that will aid institutions that cause animal suffering.