Oh, come on.  I'm not aware of any Monarch being elected or impeached by Parliament (without radically changing the nature of the country's political system.)

There have also been republics with institutions not very different from monarchies. The English in their republican phase tried having a non-hereditary chief executive. Under the Instrument of Government (1653) the Council of State elected a Lord Protector for life. Later on in the Commonwealth the Lord Protector was given power to nominate an heir, which Oliver Cromwell did by selecting his eldest surviving son. The term of office of Lord Protector Richard Cromwell was short and unhappy.

Oliver Cromwell did not have the pretensions of the Stuart Kings to be an absolute monarch, but his practical powers were much the same. He also had the benefit of a more efficient state and a disciplined regular army to ensure that his power could be exercised more effectively.

The point I am trying to make is that when a country forms a new government it tends to be similar to the old one. The differences are either cosmetic or reactions to some aspects of the old system (absolute King and Lord Protector in seventeenth century England; constitutional monarch and President subject to checks and balances in eighteenth century America).

The evolution of Parliamentary government in nineteenth and twentieth century Europe, gradually made the head of government more important than the Chief of State.

There is a modern tendency, which France gave institutional form to by giving additional power to the Chief of State, to emphasise one person (usually the Head of government) more than the government as a whole or the Parliament.

The problem with the modern British executive is not the Monarch but the Prime Minister; whose power has grown, is growing and ought to be diminished.

Ceremonial Presidents, in contrast, can't be impeached (nor have the opportunity to give reason for), and are expected to be impartial once ascending to the job (just like constitutional Monarchs). The veto power (if they have it) is their sole 'real' power, though usually conditional: e.g. a Presidential veto effectively means that the law is sent to constitutional court or back to Parliament for a second vote.

Another thing the others haven't mentioned that ceremonial Presidents are usually (but not everywhere) chosen by Parliaments (joint sessions of two houses if there is a two-house Parliament), not popular vote. Proponets see the thus limited (and Monarch-like) legitimacy as something positive: the President would be less likely to want to usurp the PM's policy authority. View it as checks and balances.

_{*Lunatic*, n.
One whose delusions are out of fashion.}

US system with executive President:

(British, Dutch, Scandinavian, Spanish) parliamentary democracy:

Parliamentary democracy with Parliament-elected ceremonial President (Germany, Italy, Hungary etc.):

Parliamentary democracy with popularly elected weak ceremonial President (Portugal, Slovakia etc.):

Parliamentary democracy with both PM and strong (some executive powers) President (France, Poland, Russia):

_{*Lunatic*, n.
One whose delusions are out of fashion.}