A lawmaker and a representative are basically the same things under most government formats that have democratic values. Lawmakers are responsible for making laws. The people elect them for a specific period of time under most government formats that involve freedom and democracy. If the government is a pure democracy the lawmaker could be any citizen of the government and might not be elected.
In our system of government, we have what is known as a democratic republic. We elect people to represent us in government. These people are called representatives because they are supposed to represent the will of the majority of people that elected these people to office. They aren’t supposed to do what they, personally, think should be done. They are to do what the majority of their constituents wants them to do. Our representatives are also involved in the process of making laws. Based on the wishes of the people they represent, they should vote for or against a bill. They may also propose a law based on what their constituents are asking them to do. Having a government system where people elect representatives to represent them makes it much easier to do things instead of having to involve all of the citizens of the country in making decisions about laws and other issues.