Experimental details are very, very important: they are used to ensure that results are reproducible, or at least that is the aim.
Let's talk about this specific example of suppliers and sources. One might naively think that a chemical purchased from supplier X is the same as the chemical from supplier Y. In an ideal world, that would be how it is. Unfortunately, this isn't the case, due to various considerations e.g. method of preparation, packaging, and so on. Therefore, even though the majority of the compound may be the same, there can be different impurities. And unfortunately, these impurities can lead to reactions working or failing. Many experimental chemists will personally know of cases where a reaction doesn't work until a reagent from a different company is used, or a reaction that stops working after buying a different bottle of a reagent.
There are many examples of this in the literature, but I'll choose the first one that came to mind, which is a reasonably recent paper from my old group (open access):
Mekareeya, A.; Walker, P. R.; Couce-Rios, A.; Campbell, C. D.; Steven, A.; Paton, R. S.; Anderson, E. A. Mechanistic Insight into Palladium-Catalyzed Cycloisomerization: A Combined Experimental and Theoretical Study. J. Am. Chem. Soc. 2017, 139 (29), 10104–10114. DOI: 10.1021/jacs.7b05436.
During these investigations, we also uncovered a critical dependence of the stereochemical outcome of the reaction on the batch of Pd(OAc)2 employed as catalyst. This discovery was made through the chance purchase of Pd(OAc)2 from a different supplier, which led to an unexpected
ratio of enamide alkene geometries ((Z):(E) = 80:20), rather than the typical ratio of ∼97:3. Screening of further samples of Pd(OAc)2 gave variable results, the most extreme being a reversal of stereoselectivity to 40:60 in favor of the (E)-isomer.
The exact supplier isn't named, but 1H NMR spectra of various batches are given and the performance in the reaction can be correlated with the exact form of Pd(II) in the catalyst, which ensures that future readers are aware of what to look out for. This is very important for good science, because what's the point of publishing your fancy new method if nobody can replicate it?
Of course, this doesn't always happen. However, it's far better to include the information, just in case it is needed.
As mentioned by Waylander in the comments, commercial samples of samarium diiodide (SmI2) also have a reputation for being inconsistent: see e.g. this report by Szostak, M.; Spain, M.; Procter, D. J. J. Org. Chem. 2012, 77 (7), 3049–3059. DOI: 10.1021/jo300135v.