Both of these bills of rights are meant to ensure that there are certain rights that are retained by the people and upon which the government cannot infringe. However, the American Bill of Rights is specifically concerned with issues of civil liberties whereas the English Bill of Rights is also concerned with what we would call separation of powers and with the way in which their government is set up.
The American Bill of Rights is specifically aimed at the individual civil liberties enjoyed by the people. It protects the freedom of speech, press, and religion. It protects the right to bear arms. It gives people the right to a jury trial with representation by counsel. It prevents the government from conducting unreasonable searches and seizures and from using cruel and unusual punishments. All of these are about civil liberties and do not have anything to do with how the government is set up.
The English Bill of Rights has some provisions that are similar to this. It guarantees the right to petition the government and the right to bear arms. It forbids cruel and unusual punishment and it forbids punishments without what we would call the due process of law. However, it also has other provisions that are more about the setup of the government. It says, for example, that the monarch may not tax without consent of Parliament. It says that the monarch may not interfere with elections to Parliament. These are issues of separation of powers, not civil liberties.
Thus, both bills of rights are meant to protect the people and limit the government, but the English Bill of Rights includes both provisions for civil liberties and provisions regarding the setup of the government. The US Bill of Rights, by contrast, is solely concerned with civil liberties.