Difference between 'surplus' and 'welfare' I see these two terms used interchangeably but I have the feeling they are not exactly the same. My gut feeling tells me this:

1)
Surplus is the additional aggregate utility from the existence of one market, while welfare is the additional aggregate utility from all existent markets

$$\implies  [ W(u_1,u_2,\dots,u_N) \ \mathrm{with} \  u_i=u(c_1,...,c_M) ] \gg [ S(u_1,u_2,...,u_N) \ \mathrm{with} \ u_i = u(c_i) ]$$

Change in surplus is thus an approximation of the aggregate gain in welfare, and is only equivalent under the conditions that: policy changes do mainly affect one market and consumers’ utility is assumed to be quasi-linear in the good at focus.

2) Or a second idea/notion, surplus is only the difference between supply and demand curve (i.e. it entails all aspects of the good that can be monetized), while welfare also includes non-monetizable aspects of a good.

Any thoughts on this?
Surplus and welfare are different concepts, but not for the reasons you state, although there are elements of validity in both (1) and (2).  They may, however, be used interchangeably in certain contexts and where certain conditions are met.

Both “surplus” and “welfare” are terms from ordinary language that in economics are used in more precise senses.  "Surplus" has several senses in economics, and is commonly clarified by the words accompanying it: (in microeconomics) "consumer surplus", "producer surplus" and their sum "total surplus", (in macroeconomics) "budget surplus (of a government)" and (in Marxian economics) "surplus value".  Where it is used alone, the context will usually make clear which of these is meant: the question appears to be concerned with the microeconomic senses.

Consumer surplus and producer surplus are normally defined in terms of supply and demand curves in a single market.  Such curves can in principle be estimated from observations of supplier and consumer behaviour, without reference to utility.  So although consumer surplus can be related to utility via marginal utility theory, utility is not needed in its definition and, as Steven Landsburg points out, surplus should be measured in monetary units, not units of utility.  Welfare, equally, is often related to utility at a theoretical level, but in applied work is often estimated without reference to utility and measured in monetary units.

“Welfare” in economics is used both in macroeconomics, where there have been various attempts to identify an appropriate set of adjustments to GDP to provide a suitable measure of overall welfare in an economy, and in microeconomics, where the focus is on welfare in a single market or a few related markets.   But these are not entirely different senses of the term.  They both retain from ordinary language the essential idea that welfare is good - something policy should aim to increase - and in this respect there is an important difference, even in microeconomics, between surplus and welfare.  “Surplus” is a purely descriptive concept (a term of positive economics), whereas “welfare” is – to borrow a little from philosophy – a thick concept, one that is both descriptive and evaluative.  Let’s consider three situations which can help to clarify our understanding of these concepts.

Firstly, consider a standard supply-demand diagram for a single product, and suppose supply increases (the whole supply curve moves to the right).  Normally, we would say that both surplus and welfare have increased.  Now suppose that the product is one which is harmful to those consuming it.  Would we still say that surplus has increased?  Yes, because “surplus” is a purely descriptive term, so we can quite consistently say that surplus has increased while thinking it might have been better if this had not occurred.  Would we still say that welfare has increased?  Quite possibly not.  Points we might want to consider before drawing a conclusion are whether consumers are aware of the harm and acting rationally, and if not whether their pleasure from consumption outweighs the harm.  This distinction between surplus and welfare can be found in debate on the regulation of smoking (here is an example).

Secondly, consider the effect of a tariff on an imported good.  The tariff raises the domestic price of the good, with resulting changes in the consumer surplus and producer surplus relating to that good. Would we say that the change in welfare is the sum of those changes?  No, because (even if we ignore effects on related markets and on exporting countries) the revenue raised by the tax is also relevant to welfare.  In a simple partial equilibrium analysis, therefore:

Welfare loss  =  Reduction in CS – Gain in PS – Increase in government revenue

This distinction between surplus and welfare can be found in (A) and (B).

Thirdly, consider a change in a market for a good subject to a production externality, say air pollution.  Here we would say that the change in welfare is not just the change in producer and consumer surplus in the particular market, but should also take account of the externality.  However, to draw a distinction between surplus and welfare on this basis would be an oversimplification since, in the economics of environmental valuation, the concept of consumer surplus is routinely applied to non-market goods.  For example (although the details are complex and the results unlikely to be very accurate) a demand curve for air quality (suitably measured) in a region might be estimated using the hedonic pricing method which uses the housing market as a surrogate for a market in air quality.  Once a demand curve has been estimated, it is possible to calculate the change in consumer surplus due to a change in air quality and include this in a measure of change in welfare. Having said that, it is questionable whether methods are available to estimate consumer surpluses for all non-market goods, so to that extent there is validity in (2).   

The conditions that would be needed for a change in total surplus in the market for one product to be an accurate measure of the change in welfare resulting from a policy change would therefore be quite stringent, including at least the following:

•  No direct effects of the policy change on other markets;

•  No effects of change in the market on other markets;

•  No harmful effects of the product on those consuming it;

•  No externalities in production or consumption of the product.

Sometimes these conditions will be met, to a reasonable approximation, and then it might be appropriate to use the terms “surplus” and “welfare” interchangeably.

Finally, a brief consideration of the idea that surplus relates to one market while welfare relates to many markets. Because “welfare” is an evaluative term, it makes sense to apply it to situations involving either one or many markets, even if in some cases it is hard to find suitable criteria to determine whether a change is an increase in welfare.  “Consumer surplus”, on the other hand is normally defined for a single market, and the path-dependency problem (see (C)) is a complication in aggregation across markets, except in special cases (this is where, if the connection is made with utility theory, quasi-linear utility is relevant).  To overcome the path-dependency problem, either compensating variation or equivalent variation are sometimes used as alternatives to (ordinary Marshallian) consumer surplus, but then there is the further problem that the theoretical income-compensated demand curves underlying these measures cannot be estimated from observations of consumer behaviour.  To that extent there is validity in (1).

References

A. Koo W W &amp; Kennedy P L (2005) International Trade and Agriculture Blackwell Publishing p 104

B. Mankiw B G &amp; Taylor M P (2006) Economics Thomson Learning pp 172-3 (a European version of the well-known textbook)

C. Johansson P-O (1991) An Introduction to Modern Welfare Economics  Cambridge University Press pp 42-52