Beginning on January 21, 2008, Kilimanjaro Airport reinstated health surveillance desks to enforce the Yellow Fever certificate requirement. All passengers entering from at risk areas must have been vaccinated. This includes passengers flying through Nairobi, even if they do not leave the airport.
It appears that the Tanzanian government is phasing in the certificate requirement and being somewhat lenient for now. But eventually, the certificate will be required for all passengers entering Tanzania, whether or not they are arriving from an infected area.
Tanzania itself is an infected area, and Yellow Fever is no joke. So you should get the vaccination regardless. It needs to be in the body 10 days before it is 100% effective and is good for 10 years. This is a good brochure on Yellow Fever by the US Center for Disease Control.