Are social goods and public goods the same thing? I am confused as to they are the same or different?
I guess it might depend on the author. Here's another classification:


  The classifications of public goods associated to agriculture identify two main categories: environmental goods and non-environmental goods (or social goods). In the first category are placed those public goods closely related to
  environmental externalities, such as farmland biodiversity, water availability and
  quality, resilience to flooding and fire, climate stability (mainly carbon storage and
  reducing greenhouse gas emission), agricultural landscape. In the second category
  are placed those public goods more related to the social dimension of agricultural
  activities, such as farm animal welfare and health, rural vitality and food security.


So under this classification social goods are non-environmental public goods.

And for some other authors, public and social goods are entirely synonymous:


  Turning to allocative problems and policies we note first, that the category of good and services provided by government are of the nature of public (or social goods). In the case of private goods, we found that efficiency in the competitive economic sense meant the maximization of individual utility alone [...]
  
  Public goods are those the consumption of which jointly by all individuals could make any one individual better off but no other individual worse off. Examples of public or social goods are public education, hightways, water and power, the postal system and so on.


There also exists some economic dictionaries, which might reflect the wider usage of the terms.

The Oxford one, in its first edition only defines public good:


  public goods Goods or services which, if they are provided at all, are open 
  to use by all members of society. Examples include defence, law and order, 
  and public parks and monuments. As nobody can be excluded from using 
  them, public goods cannot be provided for private profit. Public goods can 
  be and frequently are provided privately, by individuals and voluntary 
  organizations, from motives of altruism or ostentation. Really expensive 
  public goods such as defence are necessarily provided by government bodies, 
  which alone have the power to raise the taxes needed to pay for them. 


"Social good" is not defined in that one. There's a newer 3rd ed online of that dictionary, which also doesn't seem to define social good.

Likewise, The Economist defines only public (but not "social") good 


  Public goods
  
  Things that can be consumed by everybody in a society, or nobody at all. They have three characteristics. They are:
  
  non-rival - one person consuming them does not stop another person consuming them;
  
  non-excludable - if one person can consume them, it is impossible to stop another person consuming them;
  
  non-rejectable - people cannot choose not to consume them even if they want to.
  
  Examples include clean air, a national defence system and the judiciary. The combination of non-rivalry and non-excludability means that it can be hard to get people to pay to consume them, so they might not be provided at all if left to MARKET FORCES. Thus public goods are regarded as an example of MARKET FAILURE, and in most countries they are provided at least in part by GOVERNMENT and paid for through compulsory TAXATION. (See also global public goods.)


They also define 


  Global public goods
  
  Public goods that cannot be provided by one country acting alone but only by the joint efforts of many (strictly, all) countries. Some economists, along with global institutions such as the UN, reckon that such goods include international law and law enforcement, a stable global financial system, an open trading system, health, peace and enviromental sustainability.


There's also a Routledge dictionary of economics, which defines (quoting 2nd ed. of that):


  public good (H4)
  A commodity or service which is available
  to everyone in a particular catchment area,
  cannot be withheld from non-payers and
  is ‘non-rival’, i.e. one person’s consumption
  does not diminish that of others. The
  main examples are national defence, sewerage,
  street lighting, lighthouses, public
  health measures such as mass vaccination,
  and scientific research. As individuals
  cannot be charged according to consumption
  for goods and services collectively
  provided, it is usual to finance their
  production by taxation. Although the
  provision of public goods was small until
  the twentieth century, writers as early as
  PETTY, SMITH and John Stuart MILL argued
  for their existence. It is not always easy to
  distinguish between a public and a private
  good as some private goods gratuitously
  benefit third parties, e.g. maintaining a
  garden in a beautiful condition confers
  pleasure on many people in the vicinity.
  See also: club good; local public good;
  mixed good; private good 
  
  social good (H4) see public good


The MIT dictionary of modern economics has quite long entries, so I'll only reproduce snippets here (from 4th ed.). For public good is says among other things:


  The extreme, or
  'polar', case of a 'pure' public good has been
  defined by Paul A. SAMUELSON as a good
  which is:
  1. non-rival in consumption
  2. has the characteristic of NONEXCLUDABILITY
  - that is, if the good is provided
  the producer is unable to prevent anyone
  from consuming it.


This source also defines "collective goods" (defined by non-excludability alone), "local public good" (public goods may depend on the chosen community), "mixed good". The entry for the latter is quite long, and says in the end


  Mixed goods are certainly more common
  than pure public goods and possibly more
  common than private goods - thus their
  analysis is of great importance. The term
  impure public good is sometimes used to
  describe mixed goods.


The MIT dictionary does not define social good.

The New Palgrave dictionary (which is more like an encyclopedia, with articles/entries contributed by different authors) mentions (in its second edition) that 


  The
  distinction between private and public or social goods arises from the mode in which benefits become
  available, i.e., rival in the one and non-rival in the other case (see Public Goods).


So that entry (which is actually on merit goods) considers the terms social and public goods synonymous. They further use the public/social goods interchangeably a few times in their entry on Musgrave. Musgrave himself wrote the "public finance" entry for that Palgrave dictionary, and it too uses social good to mean public good at one point.

Their entry for "public goods" (written by Agnar Sandmo) is basically dedicated to the Samuelsonian approach to the topic. It doesn't define or mention social goods, but it does define a "social welfare function", talks of "social marginal cost of the public good" etc. There's no entry for "social goods" in that Palgrave dictionary (even though it's 7,000+ pages long).

I think the trend from all of these is clear: "social goods" are much more idiosyncratically defined, if defined at all. Most authors who do use the term "social goods" seem to use it synonymously with public goods (unlike what Wikipedia claims, without a citation.)