Seed starter SOIL is soil that is rich in humus, has some organic matter added to it, such as peat moss, and perhaps contains some perlite for soil aeration. It is designed specifically for starting young seeds. Unless it has been sterilized, it contains billions of microbes, such as bacteria and fungi. It contains nutrients for plant growth and provides an excellent medium for starting seeds.
Seed starter SOILLESS MIX is normally a mixture of peat, vermiculite and/or perlite, with perhaps a few other non-soil ingredients included, some of which are proprietary to the vendor. Notice that in the soilless mix there is NO garden soil as we know it, and there is no microbial life. The theory behind soilless mixes is that you will add the nutrients to the water, which will aid in plant growth. Also, since there are no fungi present, the problem of the damping off fungus is reduced, if not altogether eliminated.
We have found that a mixture of 50% seed starter soil and 50% soilless mix makes an ideal growth medium for new seeds. If you use only seed starter soil, it tends to pack down too much, holds too much water, and starves the roots of the seedlings for air. If you use only a soilless mix, the young plants have difficulty holding water and tend to dry out quickly. So we like to use a combination of the two planting mediums, which has always worked well for us. It is also cheaper doing this, as the good soilless mixtures can be pricey.
As for the damping off fungus problem, if you spread a light layer of finely milled sphagnum peat moss on the top of your 50/50 mixture, it seems to prevent this disease, or at least contains it to a manageable level.