Why do airlines issue boarding passes?

You have posed a good question.

In the world of aviation, the issuance of ticket and getting into the aircraft are two different parts of the entire exercise.  Once you select your flight and date and the price of the ticket is paid, a ticket is issued.  The ticket shows the travel details like your flight, date of travel and the accommodation provided to you like the class of travel etc. name of passenger and the cost of the ticket and the conditions of carriage.  It also shows the authority who issued the ticket and the ticket number etc. 

With this document one can enter the airport and contact the check in counter from where a boarding pass with seat number is issued along with the baggage tags (if any baggage is checked in for the flight by the passenger). 

Civil aviation rules and aircraft insurance companies rules demand that the full passenger list (names of passengers traveling on a flight), cargo manifest (containing the details of cargo being flown on that flight with the Airway Bill Number, type of cargo and the weight etc.) are to be put on board a flight before departure and taken away at the destination on arrival.  Merely by issuing tickets one can not expect that all passengers thus booked on a particular flight are traveling on that flight without fail.  The airport operation and ground handling of a flight involves different agencies and all the agencies involved do not accept the ticket copy as a proof of travel.  Thus for easy coordination and to establish that  passenger is genuine, a boarding pass is issued to be shown at all the points of check by different agencies (immigration, customs and security department) and they stamp on the boarding pass to prove that the passenger has gone through their jurisdiction at the airport and complied with the existing regulations of air travel whether it is domestic or international. 

There are several cases when some of the booked passengers do not travel for reasons best known to them.  There are many personal reasons like delayed arrival at airport,  last minutes sickness, last minute changes in travel plans, some one close to the passenger has died or in a very critical condition etc. which force passengers to cancel their journey. But still their names will be available in the booking list and if the booking list is considered as the final passenger list all those who did not report for travel also find a place in the list. 

Final passenger list  is the check list to know the number of passengers  traveling on a flight and this can be achieved only by issuing boarding passes to each and every passenger, whether it is an infant or a staff of the airline traveling on free ticket.  Ninety per cent of the staff do not make a reservations as they can travel only if there are seats available in the flight.  They do hold a ticket but without any particular flight reservations.  They too are issued with boarding passes if accepted for a particular flight and their names thus appear in the final passenger list.

In case of an unfortunate incident of crash which ends up in the death of some of the passengers, the passenger list is the most important legal document to verify the names of passengers who lost their lives in order to pay compensation by the insurance companies. 

To confirm that the passenger has reported for the flight, the luggage checked in, and the stamp on the boarding pass in India means that the passenger has passed through security frisking and is ok to board. This also helps Load and Trim department to plan the flight in advance.