Does the pollen from different variety Capri figs differ? 

Likely there will be differences in the quality/quantity of pollen and thus there are specific Capri fig varieties selected to pollinate fig orchards.

But is there different types of pollen required to pollinate Smyrna figs versus pollen required to pollinate common figs?

Some caprifig types (varieties) may provide better pollen than others.
Comparisons of pollen characteristics from diff caprifigs may be done at laboratory level as well as in the field by assessment on how certain cultivars react when given pollen from a number of different sources

Read this article dealing with the subject and involving Smyrna and San Pedro cultivars in Tunisia

https://www.researchgate.net/publication/251563579_Effect_of_pollination_intensity_frequency_and_pollen_source_on_fig_Ficus_carica_L_productivity_and_fruit_quality

Through times, in this area, Common cultivars meant for drying were mostly subject to caprification... long experience showing that this would result in a better fruit, larger, heavier, higher brix
and this was a plus for the farmers caprifying their figs  and obtaining more tonnage and higher classification grades.. thus a better price

There were no differentiation on the caprifig types... as they were used indiscriminately for all figs.. Common, Smyrna and and San  Pedro cultivars

Francisco
Portugal

Thanks Francisco for helping me understand caprification a little better!