In my experience "sub-Saharan Africa" has always been used to refer to all of Africa south of the Sahara, not just the countries immediately bordering it. In this case Sub-Saharan Africa includes at least the following countries:
South Africa
Lesotho
Swaziland
Mozambique
Zimbabwe
Botswana
Namibia
Angola
Zambia
Malawi
Tanzania
The Democratic Republic of Congo (formerly Zaire)
Congo Brazzaville
Uganda
Rwanda
Burundi
Kenya
Central African Republic
Gabon
Equatorial Guinea
Cameroon
Nigeria
Benin
Togo
Ghana
Cote d'Ivoire
Liberia
Sierra Leone
Guinea
Guinea Bissau
Senegal
The Gambia

I'm not sure about places like Somalia, Ethiopia, Eritrea and Burkina Faso because, although this is shameful to admit, I'm not entirely sure where the boundary is between the Sahara and not-Sahara.

In other words, sub-Saharan Africa is all of Africa except North Africa. This might be the same in many people's minds as Arab Africa vs. black Africa (this is pure speculation on my part). I've noticed that when Afro-pessimism rears its head, it's almost always sub-Saharan Africa people are being pessimistic about. This entire, rather large region of the world is frequently stereotyped as a political and economic basket case with some nice wildlife. Do a search on just about any of the countries named above and the results will be an even mix of bad news, information to do with aid projects and tourist information. There is genuinely a lot more to it than that, but it tends not to make good copy. Roll on the African Renaissance.