From my reading in maths, I have found the distinction useful in organization and presentation of results. I find it hard to believe that Hardy would have favored a "theorem only" approach. The distinction I find useful is essentially the following:
Theorem - main result of the paper. One to three theorems per paper, unless a long paper with many sections, where one to three theorems per section may be appropriate.
Proposition - result that is used in the proofs, but which may or may not be proved in the current presentation, and for which no originality is claimed.
Lemma - technical result used in the proof of the theorem, which is claimed as original and proved, but the main interest in which lies its use in the proof of one or more theorems.
Corollary - a specialization of a just presented theorem, in terms more likely to be useful in practice, or of intuitive interest.
For example Zorn's Lemma on partially ordered chains having maximal elements is not of much interest in itself, but it is key to have that established before proving Hahn-Banach Theorem or Tychonoff's Theorem.
I believe that this categorization is actually very useful for the following reasons:
(1) It helps the reader understand the purpose of a result in the larger scheme of the presentation, and to differentiate between results that are to be identified with the paper/section/chapter as part of its raison d'etre, versus ancillary results that may be important but are only being formally identified for use, proved or not, and not claimed.
(2) If one sticks to theorems only, particularly if one is not numbering within section, then one quickly reached double digit theorems, and there is potential cognitive interference between the section numbers and the decimal notation.
I'm surprised this type of thing doesn't have a well known codification somewhere ... it probably does, but I thought I would offer these arguments in favor of the distinction since the thread consensus seems to be trending in the other direction.