Numbeo's comparison site is usually useful, and it already shows a lot of aspects you have to consider when moving between countries. Here is an example comparing Budapest, Hungary with London, UK.
It doesn't check everything though, as there might be huge gaps between what is offered by a government in one country for free, that will cost you a lot of money in another. Good example is when comparing most EU countries with the USA, as while taxes are lower, and pays are somewhat higher in the USA, you have to take a lot of extra, long-term expenses into account (like health care)
So you have to check two things: the basic, more visible expenses and the long-term, not so visible expenses. You also have to take the taxing into consideration, as you'll only have your take-home-pay to pay for these.
The basic expenses you have to check:
- going out
- car/travel costs
You also have to think about long-term expenses, like:
- Health insurance / health related costs
- Schooling / tuition costs (not only for you but also for your children)