Traditional banks do transfer money to other countries, they are regulated and they are trustworthy. The downside is that international payments and international financial transfers are not their speciality. That's why dedicated companies exist that specialise in money transfers abroad. Such companies offer a much more competitive and quicker service than banks. The companies found on FundsAbroad.com are equally regulated, as the banks.
If you need to transfer a large sum of money abroad then the benefit to your wallet of employing a transfer company is clear as they offer better exchange rates and lower fees than banks. Banks have supplementary fees that you would not expect to find with a transfer company, like handling charges and foreign exchange commissions.
Equally if you are regulalry transfering smaller amounts of money a transfer company will save you money. Some transfer companies have specialised staff to help with regular transfers (like pensions). These are employed to get you the best deals possible.
In conlusion you can use a bank to send funds abroad but a dedicated transfer company will charge you a lot less, find the best exchange rates and will be quicker. At the same time reputable, regulated transfer companies offer the same security as a bank.