I think it's a whole field of research I'm not very familiar with. Personally, I'd start looking into the history of welfare in England a la the late feudal and emerging capitalist period of social welfare known as the "poor laws", seeing how they would develop into the more familiar contemporary welfare system of the UK, then delving into U.S history of welfare during the same period, and then examining the Tsarist Russian period and the Soviet period's welfare systems. But if you want to abbreviate that a bit then simply doing a contrasting analysis of social welfare of the U.S and the U.S.S.R in whatever period you desired to learn about.

Now in terms of what I know and can extrapolate off the top of my head, social welfare as broadly defined to include education, health, and social security under a socialist or capitalist system tend to wildly differ from one another in some aspects and in others - depending on states - appear similar.

On education, nearly all countries have some form of compulsory education that tend to be state funded to a certain ages with variations being dictated by their own national standards. An example of this could be how the U.S and R.O.K has free public school from the ages 5 to 19 with university being individually funded whereas Japan and the PRC has free public education from the ages 6 to 15 with high school onwards being individually funded. (Reasons why so will have to be researched in-depth). There's also other minor variations such as the subsidization of aspects of compulsory education. With the exception of Sweden, Finland, Estonia and India the entire world has do not have universal free school lunch in compulsory education as funded by their State. This means it is left to the prerogative to every level below the State from province, to the city, to the school, etc. on the question of free/paid school lunch in compulsory education. Similar stuff can be looked into for school supplies, mandatory uniforms, education materials, etc.

On health, it's also a mixed bag. The question of analysis can simply range from the availability of universal healthcare to the level of depth of funding for aspects of it depending on the standards desired by differing States. One can look at the ratio of doctors and other medical positions to the number of citizens, how healthcare is distributed, and so forth. This is more so outside my knowledge and probably needs someone more familiar with the field to examine and explain details on.

On social security, which may be broader than the first two, range from examining unemployment, senior citizen, housing, food, or a plethora more subjects as it's as broad in scope as the society it's directed towards. One can examine how differently houselessness is addressed in differing states as houselessness and combatting it tend to fall under multiple forms of social security. A simple barometer would be analyzing whether or not the state recognizes economic rights of its citizens and seeks to pursue those rights for its citizens. On this regard I can throw in this link called "China’s Employment Policies and Strategies" By Yan DI, Research fellow of the Chinese Academy of Labour and Social Security, Ministry of Labour and Social Security, P.R.China.

I'd primarily argue the main distinction of welfare under a socialist or capitalist system is in how each system performs its duties on social security and whether or not the economic rights of the worker are the primary concern of the State or not. From that point onwards it becomes the tedious task of administration of building the socio-economic structures to address the needs of social security.