Wire wrap was devised by Bell Labs in the 1940's to address some of the several disadvantages of soldering.
First and foremost, soldering represented a reliability problem. Solder joints require a certain set of skills to effectively complete, and yet the joints themselves can easily fail if the soldering process isn’t just right – and it is difficult to measure the acceptability of a solder joint and quite messy to attempt rework of a defective one. Mechanically it is brittle, weakens with age and varies widely in strength from connection to connection. Also, soldering is prone to ‘solder splash,’ or the exfoliation of small balls of molten solder from the area of the solder joint. This can cause shorts and burn damage in adjacent electrical and electronic gear. And if the soldering iron is not properly grounded, there is a risk of electrical discharge to the costly network elements in the facility.
The other big 'cold jointing' technology, crimping was devised for the same reason.
It seems that the Apollo Guidance Computer was a wire wrap job too.