Is the tomato a very very close relative of the pepper?
There are different unrelated plants called "pepper" in English. Black pepper (species of the Piper genus), as indicated by Chris, is not related to tomato and doesn't "look alike".

I guess the OP meant actually a different plant (Capsicum genus), to which the common bell pepper (Capsicum annuum) belongs. This one is similar to tomato and they both belong to Solanaceae, the same subfamily Solanoideae, but different tribes: tomato is classified in Solaneae and Capsicum in Capsiceae.

In this study you can see, how distant phylogenetically the two genera are. Currently accepted classification generally reflects phylogeny, and there are three more tribes that are closer to Capsiceae, to which Solaneae is nevertheless a close relative.

The most striking morphological difference between Capsicum and Solanum is that the first one is characterized by hollow fruits, while tomato and other Solanum species have fleshy berry-like fruits.

EDIT: It's actually not a difficult task for one's imagination to see that the two types of fruit are structurally very similar. In reality a pepper-structured tomato cultivar is known, and even quantitative trait loci underlying this morphological trait were already found.