I have done a little research into this kind of stuff.
Offshore companies exist mostly to gain wealthy US customers primarily concerned with the risk of litigation, which is an increasing cost of doing business in both the USA and now the UK. People (spouses, customers, staff) will sue over anything!
They can also take customers from other parts of the world. For example if you want to be fully free of the IRS shackles (legally, mind!) you would have to renounce your US citizenship and live abroad which is pretty final. The US government does not recognise dual nationality (especially NOT for tax purposes!).
Things are a little different in the UK. Basically if you are going to be away for 5 years or more you no longer have to report any non UK income. You dont lose your citizenship if you want to come back after 5 years.
Probably some of their biggest customers though are the many rich arabs we have in London. They come here to gain "non-domicile" status. This basically means that you can send all your earnings abroad and only pay tax on what you bring into the country to live on. You are therefore building up a huge offshore tax-free nest egg. Hence the need for companies like Panama. The bad news (for me anyway!) is that this only applies to people who do not have UK citizenship. For the rest of us, and for ordinary Americans too, offshore anything is probably pretty much a waste of time.
On the plus side you will not get a better deal for brokerage service outside the States. Trust me, the grass is greener your side - us foreigners come to YOU for the best deals!!!