Very late to the breadboard vs wire-wrapping topic ... but still.
I always decide which option to use based on two factors: Is it analog or digital? And is the circuit experimental or final?
Analog vs digital circuits:
Breadboard can be used for both analog and digital work at modest frequencies (up to a few MHz).
Wire-wrapping analog circuits is not so recommended. The leads of resistors, capacitors, transistors ... are mostly round so a dab of solder is needed (see Reliability section below). So possible but breadboards are easier for this.
For digital work, wire-wrapping is perfect. At least 5 times higher frequencies (up to 20 MHz) and even more with careful routing and attention to other details (same applies to breadboards of course).
Experiments vs implement final design:
If I am experimenting, testing circuit ideas, trying proof-of-concepts, understanding other people's designs, generally "playing around", I will go with breadboards because wiring changes and constant doing/undoing of wires is much faster and easier and convenient in breadboards.
But if the design is (nearly) complete and final, e.g. implementing an existing and tested design, wire-wrapping is no doubt the way to go. After a learning curve of a few days, wire-wrapping will take only around 25% to 50% longer. As an example: the 65816 SBC in pics 1 and 2, took me around 15 hours to complete (3 evenings of 5 after-work hours).
Costs:
One-off tool costs are: 1) manual wrapping/unwrapping combo tool with built-in stripper (looks like screwdriver) and 2) wire cutter. Total less than $20.
Recurring consumables: 5 color reels of 100ft of "proper" kynar AWG30 wire are $50 but they will last you for several projects. I also bought 8-in-1 reels of non-kynar wires for less than $10. So wires are similar in price to 22AWG breadboard wires.
Finally and most importantly: the IC sockets. They are special. They are not always easy to get. And they are expensive. Even several times more than the sockets with turned pins. The wire-wrap sockets have long pins and MUST have square cross-section. Because that is where the reliability comes from. You can however (at the expensive of time) make cheap DIY versions by soldering square male header pins next to regular sockets to achieve the same (see pic 3).
Reliability:
For me, its rock-solid reliability is probably the main reason why I use it for "final" designs. Your design/circuit will not fail because of a loose wire or bad connection. Run your fingers over the wires, drop your board on the floor, throw it during transport ... Just apply power and you are back in business. I know. I brought my wire-wrapped version of Ben Eater's "8 bit breadboard computer" (see pic 4) from Australia to Europe and threw it in my suitcase along with my clothes. Powered up 1st time upon arrival. Try that with a breadboard
.
Also note how much more dense the design can be: the 8"x12" board can accommodates nearly 100 standard TTL/CMOS ICs. And that is with some real-estate taken up by the DIY sockets. Use proper ones could have brought the total count above 100.
And it is as reliable as having a PCB made.