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I have never used a Mummy type sleeping bag, rather I have hardly used a sleeping bag. I am afraid, I am more of a carry-mat, sleeping-pad and a blanket guy.
It never occurred to my mind that What must be the advantage of a Mummy type sleeping bag over a Blanket type? over even vice-versa if it is so?
Basically it boils down to "less weight for the same insulation" in favour of the mummy type for two reasons:
In a blanket type bag you have more or less two insulating sheets, one on top and one below your body. For a mummy type maybe a bit more than the area of one of those sheets would be enough to wrap your whole body. This already reduces weight and pack size.
Above that, a mummy type bag wraps your body quite closely, therefore leaving only a small air volume that has to be warmed up. The blanket type has way more "dead volume" that you have to heat and keep warm with your body's heat. To get the same insulation, a blanket bag needs more loft material, which makes it heavier and bulkier.
Another feature of mummies is a yoke or heat collar and a hood. The yoke closes around your neck and prevents warm air from escaping from the bag and both, the yoke and the hood can be closed by drawstrings, so that everything but the face is snuggly enclosed by the bag. A blanket type bag can in contrary not be closed densely at its open end, which allows the warm air to escape quite easily.
All those factors might be not of great importance in warm climates where you don't need high insulation but only something to cover your body at night. As you get to climates where the sleeping bag is necessary to prevent hypothermia, a blanket type bag that provides the insulation needed will soon be way to heavy and bulky.
Just as a side note: if you go to very cold climates (winter mountaineering, arctic tours), everything is done to prevent heat loss, therefore some of the bags designed for those climates have only a half length zipper or even no zipper at all to prevent cold bridges.
The major drawback of the mummy bag is its tightness. Rather "active" sleepers that move a lot during the night or like to take other sleeping positions than just lying straight (legs spread, fetal curl or similar) might feel a bit less cramped in a blanket type bag. But some manufacturers also have mummy bags that have a more flexible and expansible material in the middle part to allow for leg movement to some extent.
TL;DR: Whenever it gets cold enough that insulation is a critical factor, a mummy has less weight and pack size for the same insulation.
My answer will assume we are talking about a quilt. You can get down quilts that are made of exactly the same material as a mummy bag (Pertex Quantum face fabrics with 800+ fill down). Most quilt users that are going into colder climes will add a down balaclava to keep their head warm. Futhermore, quilts are often cut in the same shape as a mummy bag (on top), specifically to prevent creating areas of dead air inside the bag. Expect any good down quilt to look very similar to a mummy bag until you look under it. They will also have a yolk as @AM_Hawlk mentioned.
Quilt Summary
For me, the cons kill quilts - I like zipping into my mummy bag and not having to worry about lying perfectly still :) I think in summer trips, they are a pretty awesome idea... I always end up half out of my mummy bag anyways, no matter how light it is, so why not?
