In one liner Progress tax is the tax with the increasing tax rate on different slabs and Proportional tax is of fixes rate tax to all slabs. 

If the tax rate increases with gradual increase on income it is progressive tax. There are different slabs of incomes and proportional tax rates are defined by tax authorities.  It works as there are three slabs 0-1000, 1000-50000 and 50001-Above with a respective tax rate of 5%, 7% and 10%, the tax calculation for a person with income of 35000 will be on 5% for first 10000 and 7% for balance 25000. 

This tax is considered to be most justified, as low income individuals have to pay less tax and more tax paid by high income people. Implementation is easy with less resistance by public. Progressive tax helps to decrease the inequality of wealth, which leads to number of social benefits.  It is also a view that rich people have more assets e.g., land, businesses, assets etc. which took more government infrastructure to taken care of these assets. So it is expected that rich should pay more tax as compare to poor people. 

Unlike progressive tax a fixed tax rate is charged for all income individuals in proportional tax rate. For instance if the tax rate is set to 10% it would be remain same for a person whose taxable in income is $10000 and also for a person whose income is $70000. 

[adsense1]This system is not used in many developed economies and there is a debate that if this is a fair system or not.  Rate of tax is adjusted by keeping in view rich and poor people of a country, in this way poor will be suffered form a tax rate that brackets them with rich people, and also state suffers to collect revenue because greater tax revenue cannot be collected as high income individuals are paying the tax with general rate.

In most economies of the world progressive tax is being used. There are some countries of world, which are using proportional tax rate as flat tax or with some modifications in proportional tax. Marginal flat tax, flat tax with deductions and hall-rabushka flat taxes are also discusses in this domain.  Proportional tax rate follows the communism school of thought but in modern communist societies are also changing their tax structure. 
Progressive tax system has an impact of capitalism. It is said to be best system where state and public equally benefited of this method. The low income people have less tax burden whereas high income individuals pay more for contributed to the government’s earning.