if your contract is through a an agency (let's say A), i.e. A is the umbrella company, this basically means you are an employee of A. A will find new contract positions for you, take care of some tax obligations for you as the employer. Because of these services A provides you, A will take certain percentage out of your pay as a fee. One of the advantages of this arrangement is you are likely to find new positions fairly easily if you want to.
If you have your own company, and sign your contract direcly, it basically means that you are an owner of a business and at the same time an employee of your own company. You need to take care all relevant taxes on your own, that would include business related taxes and indiviaul income taxes, tax as an emploer as well as an employee. Because there isn't a middleman, you will be able to take full control of your money. Also because there are more tax obligations involved, it is usually the case that people work in this arrangement charge higher hourly rate than the first arrangement. One of the advantages of this arrangement is that a different countries have different policies, very often, in favour of small businesses. You may be able to work around the system and maximise your gain. If one can develop a long term good relationship with a couple of companies, this arrangement is very often prefered by more experienced people.