They are the same power - the difference in terminology is due to whether there is a Will being probated.
"Letters Testamentary" are given to personal representatives (called 'executors') of a probate estate who were named as executor in the decedent's will.
"Letters of Administration" are given to personal representatives (called 'administrators') of a probate estate where the decedent did not have a will.
One small caveat to the above: Letters of administration c.t.a. refer to letters granted to a person not named in the will as an executor, but who is serving as such, as the named executors cannot or will not act. Letters of administration d.b.n. refer to the letters granted to a substitute executor or administrator.
As an add-on, the powers may be quite different. The powers for letters of administration are strictly statutory. The powers under letters testamentary are defined by the will, although they may be interpreted by statute as well.
The issue, as others have stated, is whether the PR appointed by the Court is a PR named in the Will.
If Yes, then Letters Testamentary.
If No, then Letters of Administration.
Note that you can have a Will with Letters of Administration if none of the persons named as PR in the Will is able and willing to serve. In this case, the Court appoints an unnamed-in-the-Will PR & issues Letters of Administration with Will Annexed.
Isn't probate and all its history and tradition fun? Am I a minority of one?