I'm also new here, but I'll try to answer...
Tennis shoes are designed for the tennis court surfaces and are often made with more durable soles to prolong useful life. Most good tennis shoes also come with a wear warranty.
Tennis shoes seem to have better lateral stability than running shoes, and even more than court shoes meant for basketball.
You won't find them at many main stream shoe stores. For example, the nearest Nike outlet here only has one tennis shoe last I checked.