The New York State Archives' Probate Record Pathfinder says:
Probate is the court process following a person's death that includes
- proving the authenticity of the deceased person's will
- appointing someone to handle the deceased person's affairs
- identifying and inventorying the deceased person's property, in some cases
- paying debts and taxes
- identifying and notifying heirs
- distributing the deceased person's property according to the will or, if there is no will, according to law
Probate records include wills, estate inventories, letters of
administration, and other documents relating to the administration and
settlement of deceased persons’ estates. These records contain
information on the property of decedents, the identity and
relationships of heirs, and legal actions taken to prove wills and
The Law Dictionary (Featuring Black's Law Dictionary Free Online Legal Dictionary 2nd Ed.), in its article What is LETTERS TESTAMENTARY? says:
Entity assignment as estate administrator or executor by this court
-issued document. Necessary if no will (intestate) for the deceased's
estate. Refer to Letters of Administration.
One curious thing about the newspaper article linked to in the question is that Ida O. Vannoske's estate is not the only one listed that apparently has both a will, and someone being appointed as administrator by means of these letters. Usually estates were either Testate (where the person left a will) or Intestate (where there is no will); letters of administration usually are granted for probates where no will exists.
However, Judy D. Russell, The Legal Genealogist, has shown once again that "It depends." in the post The testate intestacy, with an example of a case where a will is left which doesn't cover the whole estate. (For an example of the term probate being not quite what we think, see Term of the Day: probate.) Both these posts have citations to the terms in Black's Law Dictionary.
Nolo Press' site Nolo's Plain-English Law Dictionary defines letters testamentary as follows:
The document a probate court issues to an executor (personal
representative) of a deceased person's estate, authorizing the
executor to settle the estate according to the terms of the person's
will. Banks, brokerages, and government agencies often require a
certified copy of the letters before accepting the executor's
authority to collect the deceased person's assets. (See also: letters
It could be that in modern usage, Letters Testamentary refers to the letters authorizing a person to be executor where there is a will, and letters of administration refers to the letters authorizing someone to administrate the estate where there was no will.
A professional researcher experienced in New York State records might know; without consulting a pro, you might be able to find some clues by searching for a similar list for the period before 1922, so you could cross-check those estates against the microfilms which are available on FamilySearch.