Sorry, I'm not being clear.
Imagine 100 hypothetical cervical cancer patients. Almost all of them (probably over 99 of them) will have had a persistent HPV infection.
Out of those 99, around 70 of these cases will have been caused by HPV 16 or 18. The rest will have been caused by other types of HPV.
These 70 women would have been protected by vaccination.
The jabs protect against these two strains (16 and 18) but they don't provide protection against all of the HPV types that lead to cervical cancer.
The virus basically enters cells and causes them to behave abnormally. Sometimes, this change in behaviour can lead to cancer, but not always. The screening programme is there to look for these early changes before they develop into cancer.
My point is that HPV infection is associated with virtually all cases of cervical cancer.