what are the difference between the democratic republic syst

Yup,they are the same,the former suggesting democracy has been established that deprives the monarch of any real powers,or else there would be no constitutional monarch,as in Britain.
In THEORY,the power of a British monarch is extensive.She appoints ministers,public officials,military officers,judges,ambassadors,bishops (Church of England only ) and other senior public officials.She summons and dissolves Parliaments,assents to bills,issues orders,charters,patents and other official instruments,and is the authority by which many other acts of state are performed.She is the head of the executive and judiciary,part of the legislature,and head of the armed forces.In theory,she is more powerful in her realm than the President of the United States is in his republic.
HOWEVER!The reality is very different.Elizabeth II is a constitutional monarch.She reigns but does not rule,and the great powers nominally vested in her are exercised in her name by others.Although she still signs papers,presides over councils,and goes through the motions of being the head of the nation,her real opportunities to act on her own discretion are very limited.In the words of the nineteenth-century economist and constitutional writer,Walter Bagehot,she has the right to be consulted (by her ministers),to advise and to warn,but there is no obligation on her ministers to take her advice or heed her warnings.It is they,and particularly the Prime Minister,who decide on the orders,appointments and other executive acts to which the Queen puts her signature.