The advantages of striker-fired pistols are generally simpler operation, a reduced number of parts, and fewer protrusions on the exterior of the gun. Generally, the two most significant simplifications are a lack of external safeties, and a trigger pull that is the same for every shot. This makes the gun simpler to use, particularly for someone with little or no shooting experience, and is one of the major reasons for the widespread adoption of these pistols by law enforcement.
The advantage of hammer-equipped pistols is that most have external safeties and/or a heavy double-action first-round trigger pull, making the pistol more difficult to operate by unauthorized users. (In addition, some hammer-equipped pistols are double-action only or DAO, and have a heavy trigger pull for every shot, similar to a double-action revolver that is not hand-cocked.) In addition, many (albeit not all) hammer-equipped DA automatics have second-strike capability, i.e. the hammer will lift and hit an unfired round again if the shooter pulls the trigger a second time. Striker-fired pistols categorically cannot do this. Also, when firing in single-action mode, most hammer-equipped pistols have a much shorter, lighter, and crisper trigger pull than a striker-fired gun, which feels more refined and makes them easier to shoot with a high degree of accuracy.
AFAIK all modern striker-fired designs have polymer frames, which have the added advantage of being lighter and absorbing more recoil than a metal frame, and these pistols generally have a low bore axis for better recoil control. However, some new hammer-equipped DA/SA designs have emerged with a polymer frame and/or a lower bore axis, so this is not a universal advantage of striker-fired pistols any longer.
IMHO it's important to note that some of the advantages of striker-fired pistols- i.e. simple controls, no safeties, uniform trigger pull- are shared with traditional double-action revolvers (assuming the shooter does not manually cock the hammer for single-action fire). DA revolvers also have an additional advantage since the cylinder advances every time the trigger is pulled; this gives them second-strike capability on a fresh round without having to manually operate the slide to eject a dud. However, DA revolvers have a number of disadvantages- lower capacity of typically 5-8 rounds, typically slower reloading, a high bore axis, and generally sharper recoil.