Polyurethane sheet is considerably (about 30-40%?) better insulator than polystyrene-based styrofoam (EPS) or XPS. It's also completely water vapor tight, unlike most grades of EPS or XPS. I'd say it's about 0.024 for PU vs. 0.032 W/mK for EPS/XPS, but do check the actual product datasheets.
Polyurethane is used in all modern freezers for this reason. Widely used in house insulation as well. Often available with aluminium sheet glued on. And it's tougher and generates less crap when cut/drilled than styrofoam.
Even the price difference isn't meaningful when adjusted for better insulation and other savings from being able to use thinner structures (think about 50mm wall instead of 75mm for the same result).
So, if a standard styrofoam box is not good enough and you need to DIY and design your own and you have found very good insulation as a requirement in your application, I would definitely go for PU instead of EPS/XPS.
Some aerogels can be even better, but most actual aerogels available in the real market are only about 10-30% better than PU, with 10x the price and iffy availability.
While 0.010 W/mK aerogels do exist, the typical 0.020 W/mK aerogel is a joke, unless you need it for some other specific reason than the insulation performance - for example, temperature tolerance. I know people buy these products simply with their mental image of them being "extremely good thermal insulators", without actually checking the numbers.