There wasn't always a global economy. It used to be that oceans and vast stretches of land greatly limited interaction between distant economies, and the conditions in each would vary widely. As the world moves toward unification, the realities of individual countries have resulted in different interest rates in different areas.
Despite the fact that central banks set interest rates by decree, the rate only applies to overnight borrowing between banks and can still fluctuate widely from the target. In reality, the effective rate that people, companies and governments see is set by the market. The interest rate on Treasuries and other government debt determines the government's cost of borrowing for different periods of time. These rates then become the determinant of the rates offered by banks to consumers and businesses.
Interest is the cost of borrowing money, and the interest always depends on the creditworthiness of the borrower. Ironically, those who need the money most are required to pay the highest interest, which is how the lender tries to protect itself against default. This means that different country pay different interest rates on their debts, and these rates are used as a benchmark for other commercial lending in that country and in that currency.
Setting the benchmark Fed funds rate is the primary tool of U.S. monetary policy. Interest rates reflect demand for a currency---the greater the demand, the greater the cost of borrowing. Central banks use their power to set interest rates to influence demand for currency, which in turn can accelerate or decelerate an economy. Central bankers in different countries around the world have inherited different economic conditions and have different priorities moving forward, which leads them to set varying interest rates.
The fact that interest rates vary among countries creates the opportunity for arbitrage. That is, investors lend their money to countries whose rates are dropping, which means the value of their investors is increasing. They also look for macroeconomic trends to identify countries whose debt may be mispriced in the market, which would be viewed as a potential investment opportunity.
In response to the global financial meltdown of 2008, the U.S. Federal Reserve coordinated a simultaneous reduction of interest across several major countries around the world. Though rates may continue to vary in different economic regions around the globe, increased coordination appears to be the trend, and a movement toward a more unified global financial system seems inevitable. The European Union, which sets a single interest rate for a group of very different economies, is a key test of the viability of such a system.