Personally, I feel that individual stompboxs produce more interesting results when mixed and matched and experimented with, when compared to the built-in options/settings of multi-effects.
However, there are some very good-sounding multi-effects out there. My old Digitech RP6 has fantastic flanger patch and nice intricate delay patches. Believe it or not, even Zoom 505II multi-effects can produce a tasty distortion. It's a matter of settings. These can emulate sounds that are quite decent but if you want the distinctiveness of a tubescreamer or RAT distortion, etc, you gotta have the real deal
Also, you have to take into consideration the type of effects you'd utilise during a live situation. If your music requires you to switch lots of effects from let's say...distortion with flanger and delay to clean sound with tremolo and wah wah...then you'll likely to do the 'pedal dance' if you use stompboxes...hahah...in cases like these, multi-effects' ability to programme the combination of the effects in one stomp is really useful. Unless you're willing to invest some $$ in getting a A/B/Y looper (to select channels), a multi-effect would save all the hassle.
Though I rely solely on stompboxes during live performances, I'd chain up my multi-effect board with the stompboxes to fine tune the sound during recordings.
Whichever you use, just trust your ears!
P.s the downside to using stompboxes is the weight! the damn weight! *lol* Oh, and the need for a dedicated power supply...