"Mushroom Compost" is a product designed as a growth medium for mushrooms. However some people use it like other composts.
What is Mushroom Compost?
(From OSU Extension)
"Often sold at landscape supply houses, mushroom compost can help
amend garden soil, but should be used with caution, according to John
Hart, soil scientist with the Oregon State University Extension
Service. Mushroom compost is rich in soluble salts and other nutrients
and can kill germinating seeds and harm salt-sensitive plants
including rhododendrons and azaleas. The recipe for mushroom compost
varies from company to company, but can include composted wheat or rye
straw, peat moss, used horse bedding straw, chicken manure, cottonseed
or canola meal, grape crushings from wineries, soybean meal, potash,
gypsum, urea, ammonium nitrate and lime. The compost ingredients are
weighed out, then mixed in."
I have seen the odd reference to Spent mushroom compost being referred to as simply "mushroom compost". Ideally, the compost you buy should either be in a fully labeled bag that indicates what it is and what you should use it for, or be offered in bulk by an individual who can tell you all that and more. If this information is unavailable, I wouldn't buy the product. Pathogens can be present in compost and often the "organic" label is tacked on only to increase the price even if it's not an approved organic fertilizer.
Also keep in mind that spent mushroom compost isn't a great fertilizer. It should be used as soil conditioner, helping to build up your organics increasing the effectiveness of other fertilizers you use.
What is "Spent Mushroom Compost"?
(From PSU Extension)
"Spent mushroom substrate still has some nutrients available for the
mushroom; however, it is more economical to replace the substrate and
start a new crop. Before removing the spent substrate from the
mushroom house, the grower "pasteurizes" it with steam to kill any
pests or pathogens that may be present in the substrate and casing.
This final pasteurization kills weed seeds, insects, and organisms
that may cause mushroom diseases. Users may consider spent substrate
clean of weed seeds and insects."