The term was mainly introduced by Shoshana Zuboff in her book "The Age of Surveillance Capitalism" [0]. She was describing the way in which organisations harvest personal data to fabricate prediction products than can be traded.
She isn't saying "if you aren't paying for a product then you are the product". She is saying that our clicks are the raw material from which Surveillance Capitalists create a product they can sell. E.g. Facebook use our interactions and meta data to build a rich social graph, and when an advertiser wants a list of people who meet certain criteria, Facebook can then provide them with a list.
This is different to the reasons that governments typically conduct surveillance.
It's not that fewer problems happen, it's that different problems happen. The 'classic' problem of government surveillance is that the governments are able (and too often willing) to resort to force.
A distinct problem of surveillance capitalism is that someone in the business of selling surveillance is incentivized to invent new markets to sell their product to and new reasons for their customers to buy. Furthermore, if they're successful, that gives everyone else an incentive to copy them.
Inside a dystopian totalitarian government, a ministry of surveillance might push to increase its own influence, but it would never tolerate competitors.
Surveillance capitalism is a superset of regular government surveillance. Most governments have access to the databases the capitalists gathered, even though it would be illegal for the government to create such a database.
“Universal surveillance” is a correct, but less precise, description of the current situation.
Do fewer problems happen when a government does it? Does it matter if the government is more capitalist, socialist, etc?