Councils have provided local authority housing via dedicated departments for about a century. HAs mainly sprang up from 60s/ 70s onwards (although some plilanthropic housing trusts such as Peabody date from the Victoria era) and were non-profits dedicated to provision of affordable housing and nothing else.
Historically, therefore council housing stock has been older on average than HA homes with fewer mod cons than HA. HAs have continued to build new homes and in recent decades councils haven't. Although now some councils are building new homes and some HAs are buying out council housing, so the age of stick is more mixed. There have also been been big programmes to put double glazing, modern kitchen and bathrooms into council properties.
Council rents used to be a bit cheaper than HA although the two have gradually been brought into line.
So it used to be that HA tenants paid a bit more for more modern property on average.
All council tenancies in England have RTB. HA homes built or bought after 97 have RTA (smaller discount).
There seems to be a general feeling among tenants that councils are more hands-off LLs while HAs more actively manage, check up on things etc.
Council tenants have secure tenancies, HA tenants have assured tenancies, both are indefinite with a lot of security although council tenancies are slightly more secure on a few small points (slightly fewer grounds to evict). Increasingly, however, new tenants of both are being given periodic tenancies of 5 years.
So in summary, the differences are fading. The big difference is right to buy and even that is about to be voluntarily rolled out to a number of HAs.