In the US, jails are generally run by the county sheriff and may hold people who have not been convicted of any crime, as they are awaiting trial. Those who are convicted and sent to jail are generally convicted of misdemeanors and have less than a year to serve. Jails generally have far fewer facilities, especially as regards recreation, hobby, library, rehab, education, and chapel services. Most jail inmates spend the vast majority of their time in their cells. Jails range from tiny facilities capable of handling a dozen or so people in tiny counties to huuuuge facilities which serve large cities.
Prisons are run by the individual US states and the Feds also have their own prison system. Folks in prison are generally convicted of felonies and are doing more than a year. Most states have more than one prison, and they range from maximum to medium to minimum security institutions, with some ‘supermax’ prisons in some states also. Inmates in the supermax prisons don’t often leave their cells, but as the security classification lessens, inmates have more opportunities to be out of their cells doing jobs, recreation, study, etc.
In my experience, most inmates in Wisconsin prefer to get from jail to prison as quick as possible, as prison offers more opportunities to do things besides just sit.
The above description is a gross generalization of how the incarceration system in the US works, and many many exceptions exist.