What you "get" at both private and municipal albergues is a bed with a sheet on it, and possibly a blanket, if you're lucky.
You get a shower, usually hot, but not always.
You may or may not get kitchen privileges.
In a municipal, you are less likely to be allowed in if you use pack transport.
A private may be cleaner, but not always.
A private may have fewer beds per room, but not always.
You can usually book ahead at a private.
You cannot book ahead at most municipals.
You may or may not get a meal at either.
Both municipal and privates will almost always provide you with a sink with cold water for washing clothes and a line to dry.
You must provide your own cold water washing soap - bar soap is my preference - something like Fels Naptha.
Privates may have a washing machine, in which case soap is provided.
In a municipal, (and some privates) you will be greeted by a hospitalero who may or may not be a volunteer,
and who may or may not be kind or grouchy.
Prices range greatly from "donativo" which, as someone else mentioned does NOT mean "free."
What it means is that you pay what you can honestly afford.
Please be generous.
The money you give today often feeds the pilgrims that show up tomorrow.
If I'm getting a mat on the floor, I pay €6. If they're feeding me I pay €8-10.
I love staying in the parochial refuges, such as the one in Viana.
Though it is donativo, I will pay €10 there because they treat me so kindly and serve a nice dinner.
Municipals tend to be less expensive, but for me are often preferred.
For instance, I prefer the clean municipal in Astorga to a private I know of which is constantly bedbug ridden.
If you book a private ROOM, not albergue, you will most likely get linens, towels included.
You will pay more, but you will not sleep in a room with 4 to 50 other pilgrims.
On the other hand you will miss the wonderful experience of community.