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Answer: There are two possible reasons to forbid using steel wool to wash dishes on Shabbat.
The gemara (Shabbat 50a) says that one may generally scrub utensils thoroughly on Shabbat, except that one may not scrub silver with an abrasive material called gartekun because that will definitely smooth out the surface. (Cleaning well is permitted, whereas smoothing a surface is included in the melacha of memachek). The Shulchan Aruch (Orach Chayim 323:9) paskens this gemara and highlights the fact that silver is a relatively soft metal. While the Mishna Berura (323:39) says that for other, harder materials even a gartekun can be used, some say that steel wool is worse, as it is meant to smooth out even the surface of steel (Shemirat Shabbat K’hilchata 12:10; Dirshu 323:41). Some Acharonim (see K’tzot Hashulchan 146:(126)) claim that there are also problems of uvdin d’chol (weekday-like activity) in this scrubbing.

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The bigger problem is the similarity to a sponge, which may not be used on Shabbat because of sechita (squeezing out) of absorbent objects. (There is a machloket as to the melacha to which this belongs – see Orchot Shabbat, I, p. 399). One can argue cogently that steel wool, made up of thin metal strands, is not an absorbent object, and just as the gemara (Shabbat 128b) says there is no sechita of hair, so too there should be no sechita of steel wool. However, we generally assume that there is a Rabbinic prohibition to squeeze hair (Mishna Berura 326:25). This may also apply to other non-absorbent materials (see Orchot Shabbat 13:56).
The question is where to draw the line. The following is probably the basic guideline that most people knowledgeable about the halachot of Shabbat keep. If the non-absorbent elements of the material lie together in very close proximity, it is prohibited to squeeze the liquid between them, as this is in many ways equivalent to classic sechita (Shemirat Shabbat K’hilchata 12:15). For this reason, specially made "Shabbat sponges" are not only made out of hard, nonabsorbent materials such as plastic, but the strands are somewhat spread out. Steel wool does not meet these standards, and this is particularly a problem when it is desired that the liquid in between the strands (soapy water) come out to use for the washing (see Mishna Berura 320:55).
Therefore, poskim generally do not allow washing dishes with steel wool (Shemirat Shabbat K’hilchata 12:10; Orchot Shabbat 13:58). Nowadays, there are effective alternatives, broadly accepted for Shabbat use. However, since the idea that there is sechita on non-absorbent materials and its exact parameters are not trivial (see Shemirat Shabbat K’hilchata 12:(46)), the woman you refer to might have received a legitimate rabbinic leniency.
Even if one is going to use steel wool, he should be careful not to cut a piece off a roll or mass, on Shabbat (see Shulchan Aruch, OC 340:13), and if he is particular about the piece’s size, it violates the melacha of mechatech (see Rambam, Shabbat 11:7; Mishna Berura 340:41).

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