[susan53]

I think they're more or less synonymous. Eg here they'd be interchangeable :

1.With a dangerous and convicted murderer ON THE LOOSE and various alleged sightings of her, authorities hoped to quickly allay public concerns..

When the person/animal/thing is free but restricted to a certain area, then maybe on the loose is more likely - eg:

2. More than 100 prisoners are still ON THE LOOSE in 5 wings of the jail
3. Mystery drone still ON THE LOOSE at Gatwick airport.

On the loose gives prominence to the idea of "free/uncaptured" whereas at large gives more prominence to the idea "it could be anywhere". So it depends which aspect you want to emphasise.