We use both in our projects depending on the situation.
The cellulose is nice cus' its healthy for the installers, low embodied-carbon (recycled paper) and easy to install well as either denspack or loose-fill (attics). The down side is it can get damaged if it gets too wet. I've personally pulled wet cellulose out of roof cavities before and I promise its gross.... but still: overall it can and does work great in our projects. Pretty low cost too compared to other insulation types.
Mineral Wool is great as an exterior insulation (imo) cus' its super durable - can get wet and never gets degraded. Its also non-combustible which is important in some places we build (NYC). But, its pretty high embodied-energy/carbon (takes a lot of energy and some nasty chemicals to make I believe) as a product, as well as more expensive than cellulose in most cases. Also - be careful using it on the 'inside' of a home as I think most types use added-formaldehyde glues. You can get formaldehyde-free versions but they are usually special-order in my experience and you have to really push to be sure the supplier gets the right one. (oh: also - is itchy as all heck for the installers.... for whatever thats worth)
Basically: they both have their place I think, and either can work great depending on your circumstances.