Over the last decade or so, fMRI has been more widely used than PET scans. This is because fMRI has better temporal and spatial resolution compared to PET scans. Also, fMRI does not use radioactivity, while PET scans do. However, fMRI still retains some disadvantages in comparison to PET scans.
The MRI scanner, which fMRI uses, is noisy, which of course means it can serve as a distraction to a subject, and also cannot be used to study audition. Another difficulty is that with fMRI, small movements can distort the signal being measured. You cannot speak while in the scanner, for example. Also, some brain regions are susceptible to signal distortion because nearby tissue has different magnetic properties (for example, sinuses, oral cavity, ear canes, air voids, essentially). Therefore, this makes certain regions, like the temporal region, and the orbitofrontal cortex, hard to image in fMRI. Also, PET, but not fMRI, can be used to trace pathways of chemicals.