If they could hire someone full-time to do it, they would, I think. I say this because every single company I've consulted for has tried to pressure me into coming on full-time. I always politely decline, but it is exhausting sometimes. I remind myself that I'm fortunate and that I should be grateful for the opportunities.
Either they can't hire fast enough, or they can't have the employees deliver fast enough.
Anyway, I don't know if there's a clear pattern.
Let me just list a few examples:
1. Large company. New CTO wanted a proof-of-concept. Big company with slow teams and processes. He got me to build it fast, and demoed it to the board, and got the leadership team excited.
2. Startup. CTO and the 3 engineers were mobile devs. They were raising their next round and wanted a web version of their product. Ended up building a full product that got them much bigger traction than they expected.
3. Startup. CEO and the investors thought that the team needed to focus on X. CTO thought that Y will actually get them product-market-fit. CTO asked me to build Y, and the CEO agreed, to preserve optionality. Turns out the CTO was right.
Part of the reason you can get stuff done fast is precisely because you're not part of the company. The 'you should come work here full time' scenarios almost never seem to really get that point. The moment you come on 'full time', their perception of you changes, expectations of you change, and you're rarely likely to deliver what you could because of these changes.
Backoffice projects? Actually working on their main project? Maintenance of legacy systems?
Or basically, what kind of projects companies put contractors on as opposed to their employees/founders?