Both, sensitivity and specificity are mainly a matter of your primers only. If your primers work well, it is not a problem to detect even a single lonely target molecule (if it really happens to be in the PCR mix). This is the highest possible sensitivity, and you can get it with "fast" or "slow" PCRs.
"Fast" PCRs (rapid-cycle PCRs) usually have some advantage over the "slow" PCRs because the time spans in which the reaction mix is at some intermediate temperatures between annealing and extesion are short, and the time spans the reaction is at relatively low temperatures is shorter in total, what helps avoiding the formation of unspecific products. So for suboptimal primers, rapid cycling can have a positive effect on the specificity and thereby also on sensitivity (by reducing the competition of the amplification of unspecific products).
If the instrument allows rapid cyclcing, I would always prefer it (this applies to qPCR with short amplicons, for sure).
Agree with Jochen, the most noticeable feature for fast PCR is it just gives you PCR results quicker. However the recombinant enzyme for the PCR may have enchanced features vs. the traditional taq, i.e. the Ssofast enzyme from BioRad claims to be less sensitive to PCR inhibition. However it's not going to cause huge shifts in Cts which is mostly determined by template abundance.
Agree with Jochen and Can. If you use good primers and probe and length of the amplicon is 65-80 b.p. fast protocol of qPCR is quite useful. But it may be an increase in variability of results (Ct) at low concentrations the initial matrix in fast qPCR protocol. Thus, may fall the reproducibility of the reaction, especially if you use a "conventional" (cheep) Taq-poly.