I don't know about Canada, but skating schools and skating clubs are the same as far as I know.
The skating school is run by the rink. In cases where the rink is owned by a public entity (like a city parks & rec department) the government runs the skating school like they would other rec classes. In cases where the rink is private (most places not in Canada/the Northern US), the skating school is run by the rink itself.
In both cases the rink will have a full time skating director who is responsible for making sure classes run smoothly. They arrange classes, set up levels (and if they choose to arrange the levels through USFS/Skate Canada/ISI nationally), schedule ice time, hire coaches, and deal with issues. Outside of classes it is their job to approve or deny requests to coach at the rink - if something goes wrong they can ban coaches and they can stop coaches who caused trouble elsewhere from coaching at their rink in the first place.
Skating clubs are always private entities that may simply be a sports club or may be run as a nonprofit depending on funding/fundraising and the club's preferences. With the exception of the Skating Club of Boston I don't know any who own their own rink outside of the varsity skating programs in high schools/colleges. While many rinks have a single skating club present and as a result their skating skills kids tend to only join that club, there don't tend to be any official club affiliations with skating schools. In fact most clubs operate completely separately from the entire rink process beyond shared coaches and skaters - and in fact rinks can ban or cut ice time from clubs for any reason.
The skating school is run by the rink. In cases where the rink is owned by a public entity (like a city parks & rec department) the government runs the skating school like they would other rec classes. In cases where the rink is private (most places not in Canada/the Northern US), the skating school is run by the rink itself.
In both cases the rink will have a full time skating director who is responsible for making sure classes run smoothly. They arrange classes, set up levels (and if they choose to arrange the levels through USFS/Skate Canada/ISI nationally), schedule ice time, hire coaches, and deal with issues. Outside of classes it is their job to approve or deny requests to coach at the rink - if something goes wrong they can ban coaches and they can stop coaches who caused trouble elsewhere from coaching at their rink in the first place.
Skating clubs are always private entities that may simply be a sports club or may be run as a nonprofit depending on funding/fundraising and the club's preferences. With the exception of the Skating Club of Boston I don't know any who own their own rink outside of the varsity skating programs in high schools/colleges. While many rinks have a single skating club present and as a result their skating skills kids tend to only join that club, there don't tend to be any official club affiliations with skating schools. In fact most clubs operate completely separately from the entire rink process beyond shared coaches and skaters - and in fact rinks can ban or cut ice time from clubs for any reason.