In what way is the position "president" different in presidential and parliamentary democracies? Presidential would be like in the USA and parliamentary would be like the UK's system.
The US has a constitution. Most EU states also have constitutions. I believe what you mean with your question is the difference between heads of government in parliamentary democracies vs. presidential democracies.
The key difference, in a typical parliamentary democracy the parliament can replace the head of government at any time, for political reasons.
- This could be a vote of no confidence, as in "This House has no confidence in the Prime Minister." If the motion passes with a majority, the head of government must go and a new head of government must find a majority. Who gets to try differs from country to country, usually someone makes a judgement call as to who might be able to find a majority in parliament. If there is no majority, the rules might call for new parliamentary elections.
- This could be a constructive vote of no confidence, as in "This House wants John Doe as new Chancellor." Here a majority against the old head of government isn't enough, it must be a majority for a new one before the old one has to go.
Of course knowing that this is possible will alter the shape of politics. When a head of government must have the support of parliament, he or she must respect the wishes of a majority of the legislators much more than a head of government who "merely" needs to pass a budget every now and then.