I've use both approaches pretty extensively, for both bass and guitar, and prefer individual pedals. My why's:
• Multi FX almost always means all-digital. Digital is fantastic for some things, and not desirable for others, in my opinion, and for what I like. For example, I don't want my dirt, wah or delay to be digital. But digital modulation and reverb is glorious to my ears. Simulations and models of analog are not the same. YMMV
• I find that programming patches is a tedious pain in the neck. I would rather twist knobs of individual boxes than scroll through menus to tweak presets. Been there, done that many times, and to me, it gets old. And I also find that I don't like revisiting and re-tweaking the patches, so I'll tend to just live with what I set up. Also, settings are usually more complicated to change quickly on the fly. YMMV
• With individual pedals, you can get the exact tone you're looking for, for each effect. You're not limited to the algorhythm and brand of the multi. Example, for me is, I love a Fulltone OCD for dirt, a JHS Panther cub for delay, a EHX for fuzz, anda TC electronics digital pedal for reverb. that's what sounds best to me. If you have an all-in-one unit, you have to live with the models of those things they include. YMMV
• It's not as much fun as mixing and matching pedals from other manufacturers. YMMV
• Sharing a single processor for all effects usually means less processing for each one. And you can often hear the difference. Meaning, the dedicated digital reverb pedals I have used sound way huger than with more headroom than the ones that came as part of a mutli-fx "everything" pedal. YMMV
• In my experience, in general, the more effects that are combined into one unit, the less good they sound individually. Meaning, it's a bunch of mediocre effects that sound OK, or some of them sound great, and some of them are meh... but if I want the best tone for each effect, individual, hand-selected-for-the-purpose boxes are better. YMMV