Addition to Sampling with and without replacement

The difference between sampling with replacement and sampling without replacement is twofold:

1. In the former case a temporal order is essentially assumed during the sampling process. Elements {a_1, a_2, .....,a_n} are drawn one after another, and at any time during the process, the same set of elements to draw from (S, say) is available.
2. Sampling without replacement can also be described as `grabbing', taking all the elements {a_1, a_2, ...,a_n} of the sample at the same time. If done sequentially, that set has to be adapted since the element that has been drawn should be removed. If  the initial set of elements to draw from is indicated by S_0, at each draw the set  from has to be adapted in the following way: S_i = S_(i-1) \ a_i.   

Funnily enough, sampling without replacement as occurs in real life research is essentially sequential and the above formulation in 2 seems counterintuitive since there is no distinct background set S apart from the sampling frame (Hand, 2008) and the population P (Think of a clinical trial in which patients are included one after another).  

In short, the two sampling mechanisms are very different and are applied in different situations.
Sampling without replacement is the sampling procedure that is commonly used in all emperical research.
Sampling with replacement is used in resampling strategies like bootstrapping. In such cases the notion of population from which the sample is drawn, is remote.

However, sampling with replacement also occurs in real life sampling, for instance in Capture-Recapture sampling aimed at determining population size of animals in a habitat, see section 12.3.6 of Agresti's book.

References
Alan Agresti (2002). Categorical data analysis. Second edition. Hoboken, NJ: Wiley.
David J. Hand (2008). Sampling. In H. J. Adèr, G. J. Mellenbergh, & D. J. Hand, Advising on research methods: A consultant's companion, Chapter 5. Huizen, The Netherlands: Van Kessel.