- 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.
- 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.
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.