Ok, I only wonder whether there is some specific semantic difference between the two, like the difference between a convent, monastery, etc. (i.e. gender segregated) or whether one larger or smaller, or more closed or more open to outsiders, etc.
For what it's worth adding to this, I looked for some etymological information in Μπαμπινότης's Etymological Dictionary, and it looks like it's from Ancient μονή 'abiding, tarrying' > 'stopping place, station' > 'place of dwelling, monastery'. I suppose μοναστήριον (originally 'hermit's cell' > 'monastery') was an extension of the original sense.
Yes, μονή comes from the verb μένω (to stay - also used in modern Greek to say where someone lives). From μένω we also get μόνιμος which means permanent, constant, standing. The ending -ήριο(ν) is a very common one denoting a place where whatever is before the dash happens. For example, καπνίζω=to smoke, το καπνιστήριο=the place where people (are allowed to) smoke, καθαρίζω=to clean, το (στεγνο)καθαριστήριο=the (dry)cleaner's etc.
In the case of μοναστήριον the verb would be μονάζω = to live alone, to live as a monk.
As Troll has said, there is no difference between μοναστήρι and μονή, and either can refer to both monasteries and convents, so convent should be accepted as well.
In Greek the following forms are used to specify the gender of the inhabitants of a monastery:
- μοναστήρι ανδρών/γυναικών
- ανδρικό/γυναικείο μοναστήρι
- μονή ανδρών/γυναικών
- ανδρική/γυναικεία μονή