If you privately rent, you have a choice to accept or decline a property based on the decorative standard. Most private LL's know this, and have to keep their property(ies) at a standard that people are willing to pay the market rent for. You wouldn't take out a tenancy on a PR property if it had damp running down the walls, no carpets/flooring and mould growing up the walls, would you? Yet that is exactly what is expected of people in social housing.
Therefore, people in social housing often have no choice but to pay for these things themselves, or live in unsanitary conditions. You would hardly pay hundreds of pounds to carpet an entire house if you were going to be forced to leave in 2 years and you knew the carpets would be thrown in the bin at the end of that time.
No-one in Private rented has to worry about the cost of getting carpets fitted when they move in, no PR LL would get tenants if they had no flooring. Has anyone seen the price of carpets these days? So yes, people in PR do often have to move every 6 months rather than 2 yearly, but can you imagine the extra costs over, say, a ten year period, if you had to recarpet an entire property every 2 years?
Both sets of people (those in social housing and those in private rented) will have to bear removal costs, but if you are moving white goods as well, like people in social housing are, that costs more, as you may well need more trips to move your stuff.
So you would have added moving expenses of carpeting and added expenses of removing white goods.
I am currently in private rented, but have been in social housing in the past, and I can gurantee it costs just as much, if not more, to move social housing every 2 years than it does moving between private rented properties every 6 months.