Are heavy blankets effective insulation when used over drafty windows as curtains or when hung in doors to shut off an unused space?
Is it more effective to seal the windows with plastic?
Yes, heavy blankets can be effective insulation over draughty windows; or in open doorways to shut off an unused space: over both, they will hinder heat loss through convection (air currents), and over windows they will add a bit of a conduction barrier too. Castles used to hang tapestries over all the walls, partly to keep some heat it.
Ideally, they should be floor-to-ceiling, which helps cut out convection bypasses. The convection bypass of concern here is the air rotation by natural convention around insulation, the third image below. If the thermally insulating curtains (blankets) don't go floor-to-ceiling, then air will get cooled by the window, drop down to the floor, pass under the bottom of the curtain into the room, and warm air will pass from the room, over the top of the curtain, and get cooled by the external wall and window.
As for whether they're more or less effective than clear plastic film over windows, will depend on a bunch of things.
Will you keep the blankets over windows even during daytime, or will you open them? If you'll be opening them, then the clear film may be better.
Is the main heat loss through the window due to conduction or convection? Are they draughty, or are just leaking heat conducted through the glass? If it's mostly through the glass, then the clear film may be most effective: it will trap a layer of insulating air next to the glass. But if a lot of heat is going out through the window frames too, then the blankets may be better.
Ideally, do both: put the clear film over the glass, and hang floor-to-ceiling blankets over them as curtains. That way, you'll benefit from the air tightness and reduce conduction losses.
Warning: as ever, when increasing the air tightness of a room or building, remember that one of the things the draughts had been doing, was taking away excess moisture (from breathing, washing, cooking). If you don't ensure that that moisture is taken away, you risk condensation, mould, and consequential damage to property and health. So think about your ventilation strategy. Ventilation in infrequent bursts - i.e. just opening windows or doors once a day to exchange the air completely - may not be enough. Some peopel use silica gel in pouches, next to the window, to trap the moisture, to deter mould.