Mash Tun because it only costs 40 bucks and fuels the DIY in most homebrewers. And it looks cooler.
Nice pun, looks cooler...
I use both methods and I don't have a favorite yet. I started doing BIAB, since I had a burner and pot, all I needed was a bag. Here are some of the pros and cons that I've experienced.
BIAB:
Pros - No extra equipment required except a bag. You can do step mashes using direct heat to raise the mash temp, just keep the bag off the bottom of the kettle so it doesn't scorch the grains.
Cons - Depending on your outside temperature and the insulation around the pot, it might lose more heat during the mash, but that's easy to correct. My BIAB brews have turned out a little bit cloudier, maybe because there's no vorlauf. This could just be me as I don't have a false bottom on my BK.
Mash Tun:
Pros: The cooler mash tuns maintains heat pretty well. It typically allows for a larger grain bill than BIAB. I've experience clearer wort, probably due to vorlaufing. It's a little easier to manage the mash tun than a hot bag of grain.
Cons: Costs more, although it's less than $100 for the cooler and all the parts to add on the valve and braided hose. It takes up more room when stored.