Well we are only paying for the actual security which is to protect:
A) Foreign heads of states and governments coming to the wedding.
B) The British people lining the streets on the day.
The amount of people the wedding will bring in from abroad is huge, This will more than make up for the cost.
The Monarchy actually is a profit to the tax payer. There has been a long standing agreement that the Monarch surrenders the profits from the Crown Estates (£220 million last year) this far out ways the cost of the Monarchy.
The Monarch also gets no wage from the tax payer. All money is for official expenses which a President would nee anyway and in the scale of things is actually very small. I think the President of the United States cost $1.5 billion for just the election and inauguration this dwarfs the £7.9 million the Queen gets from the civil list.
The Queen does have some role in British governance all though it is very small and based completely around constitutional enacting.
The Monarchy has three roles: Ceremonial, Constitutional and Ambassadorial.
Constitutionally the Queen appoints the Prime Minister, calls and dismisses Parliament and gives Royal Assent to bills so they can become law. She also has the constitutional requirement to advise, be kept informed and warn the government of the day. This is done with regular meetings with ministers, constant updates with state papers and weekly meetings with the Prime Ministers.
Ceremonially the Queen continues to enact many traditions, gives honours recognising the work of her citizens and also has to give her ceremonial approval when the government needs to exercise some of the Royal Prerogatives such as declaring war.
As an ambassador to the country she receives foreign Heads of States/Government and ambassadors as well as undertaking overseas trips to sustain overseas relations. She also tries to act as a symbol for the country by fulfilling charity works and other activities.
If we didn't have the Monarchy then we would have a President doing the same things.