They are allowances with different aims and different 'users' on the whole.
The old age pension was set up to provide old people with a living standard that would be above the poverty line after they retired so that they had some security and could sustain reasonable food, warmth, housing and the impact that has on health. It is a life-long benefit once people reach retirement age. It was intended to stop the state paying out lots of money to deal with the costs and impact of poverty in old age as well as providing security for the elderly. Originally, when it was introduced, it was non-contributary and a low rate but the idea emerged that people would pay a NI contribution all their working life to be entitled to a reasonable pension in old age.
Beveridge identified five things that after the Second World War that would have to be beaten: poverty, disease, ignorance, squalor, and idleness. His idea was these could be tackled with a state-run insurance system, and a minimum level of living standards would be established that were not extreme or punished the long-term sick who simply could not work.
So we all pay National Insurance and in return are entitled to a welfare state with social security, an NHS, free education for all children, council-built and run housing, and pensions for the elderly.
His idea was our NI contributions gave us access to government aid for the unemployed, ill, retired or widowed, and extra payments to support children (Family Allowance). There was no means testing- which there had been before the war.
The Post War Labour Government supported Beveridge's ideas (he was a Liberal) and implemented them. However, Beveridge expected people to still save money and do the best for themselves, very much in the thinking of the British liberal tradition. The individual would wish to be responsible for working and for supporting their family. He thought of it as a British characteristic. He didn't actually think it would cost the State money because of all of the NI that would come in from contributions from workers.
JSA, or whatever title it has had in the past, was never intended to be other than a short-term allowance to tide people through the time until they found another job. It was meant to cover the bare necessities so that it did not encourage people to stay on it. Over the years this allowance has had many forms and changes of rules. However, I still think that intention exists- that it should be very basic and not a place of any comfort. Successive governments never want to be accused by the electorate of encouraging anyone to be better off not working. That is the 'idleness' of his original 5 things that had to be tackled. It is a fear many people still have of a social security system.
I think that explains, as I understand it. In the late 40s, 50s, and 60s on the whole, the government and the electorate were all in favour of developing the system. By the late 70s when Thatcher came to power there was thinking that it cost the State too much money and that it needed reigning in. Hence raises to pension ages, higher NI contributions, selling off council housing, and welfare cuts.
This has been particularly the case since 2010 when the Cameron became PM. The Conservative party wish to reduce the role and responsibility of the State but we are fast returning to the position before the Beveridge Report where the rich are increasingly rich and the poor increasingly poor- and that has huge impact on every aspect of the lives of the poor- health, living standards, housing literacy, education, life-expectancy etc. Beveridge could not forsee the incredible developments in medical treatment that would take place and the high costs associated with that, nor the collapse of traditional heavy industry and manufacturing and the impact of that on employment.