These two concepts are often not distinguished. Yet they should be.
Socialism is a form of mandatory insurance that is imposed on everyone. It prevents the emotional hardships caused by the sympathy we feel when exposed to the bad fortunes of others. For example, I feel terrible after seeing children who are visibly sick but don't have access to medical services. Thus, before admitting a person within my geographical region —that is, in my field of heightened
sympathy—, he must be sufficiently insured so his observable misfortunes distress me minimally. There are various methods available to finance this mandatory insurance, which involve both individual and collective contributions. It's an expensive but necessary fiber in the fabric of modern society.
On the other hand, the "nanny state's" usefulness stems from the high cost of making rational and educated decisions for individuals. The ineffectiveness of individual decisions becomes particularly evident when applied to infrequent choices or longterm commitments. For example, choosing to embark on a long private educational program (such as K-12) at the cost of accrued interest and with first rewards reaped in several years is a very hard choice for any person—let alone a five year old! Another good example is private medical insurance: what ailments should I be protected against? Is it reasonable to expect individuals to research the incidence, cost of treatment and discomfort of all existing diseases? Definitely not. Educating individuals to make better long term choices is necessary to have a functional complex society, even on things as trivial as using a condom during intercourse with irregular sexual partners. Both government and non-government bodies can make such recommendations, and these recommendations usually apply across national borders.
Knowing what is "the right thing to do" for their own welfare is not always sufficient for persons to follow this path, in which case legislation is required to force them in the right direction. For example, educating children is mandatory, and not a choice left to parents. These rules can often be bypassed via emigration (to poorer countries), or become stale as "the right thing to do" changes over time. They should be kept to a minimum and often revisited or reverted back to non-binding recommendations when the right choice no longer seems universal. Occidental countries are plagued with obsolete regulations kept in force by lobbyists whose industries thrive through regulatory protectionism. One blatantly obvious example of this is pharmacists who have a monopoly of retail prescription drugs sales in Canada. Why can't I buy drugs for which I have a prescription from a Mexican pharmacy? The exact same drugs are cheaper in Mexico. The direct cost of maintaining "nanny state" recommendations and regulations is quite low as compared to the cost of mandatory insurance, since it mostly involves research and dissemination of information. The indirect costs can be quite high since obsolete industries survive longer than necessary, sustained by regulatory legacy.
The socialism paragraph above intentionally omits discussing how the mandatory insurance provides direct help to some insurees, though this seems to be another key benefit of the policy. In taking private insurance contracts, each individual must weigh his tolerance of risk. The less variance one can tolerate, the more insurance one buys (at a cost, obviously). This remains a strictly personal choice, which is guided by your culture and "nanny state" recommendations and regulations. Thus the amount of insurance one buys to maximize one's own utility is not a fixed quantity. Weighing in only their personal benefits, some persons will require more insurance than is provided by the socialist mandatory insurance, and others will require less. The good news is that you can buy as much private insurance as you want when the state has forced too little insurance upon you. The bad news is that you can't buy less: you can't opt out of mandatory insurance. Buying more insurance than you need makes you worse off, in the typical case, so measuring the benefits of a sub-group of insurees (namely those that fell into adversity) is an analytic fallacy. Mandatory insurance in no way helps one's own fate; only "nanny state" recommendations and regulations on how much insurance you should buy helps you directly.
Both socialism and the "nanny state" are required to make each of us happier. "Nanny state" recommendations save me hours of research in making infrequently encountered choices. "Nanny state" recommendations and regulations protect me from bad decisions I make in striking a compromise between my immediate and longterm wellbeing. And socialism helps me out by reducing the misfortunes of others, which affect my happiness through my sense of sympathy. Until we can knock out the sympathy gene and the hyperbolic discounting gene, we are suck with high taxes and zealous regulators. Libertarians should be spending less time blogging, and more time investing in genetic research!