In my experience, there is no level of advice or type of connections Deloitte can provide that someone cheaper and/or someone more specialized can't do.
You should use Big Four when you want to make a statement with your auditor or external consultants. It's prestigious only because it shows how much money you could afford to waste. If you're a licensed bank and need to show the central bank or other regulator that your books are in good order, you should use Big Four. Not because it's automatically better, but because people who don't know better think it is.
There are good people working at Deloitte and the other Big Four. However, the best ones are ex-Big Four, who have gone on to do other things after building skills and network at Big Four for 5—10+ years.
When you work with companies like Deloitte, they are usually so careful and risk adverse that you can easily miss out on opportunities simply by being overly cautious. This can mean loss of revenue opportunity or simply spending too much money on something that goes above and beyond what's necessary to be compliant.
For those and a few other reasons, I rarely suggest someone go with Big Four. But if your risk appetite is zero or you need to impress someone, they might be the perfect choice.