The two capture much of the same spirit, but neither contains or implies the other.
Scientism seems far more restrictive than logical positivism in one sense. To point to a place between them, we can look at the younger Wittgenstein.
Wittgenstein's Tractatus lay quite close to logical positivism, and by many estimations is the source of its original motivation. But throughout his life he always disdained scientism, clinging in his youth to the form of religion despite incompatibility with its content, and insisting later "People nowadays think that scientists exist to instruct them, poets, musicians, etc. to give them pleasure. The idea that these have something to teach them - that does not occur to them." (Quoted from his personal notes in "Culture and Value" https://en.wikipedia.org/wiki/Culture_and_Value)
There is a difference between declaring a thing not to be clearly perceived or described, or even to lack 'cognitive meaning' and declaring it to consequently have little or no value. There is space for someone who values groundedness in argumentation and is not a Philistine.
In another sense, Scientism survives the death of Logical Positivism by being less specific. Given the logic of Quine, etc. and the deconstruction of much of the program of positivism, many who only wish to value what is statistically or methodologically validated do not have to adopt any specific underlying philosophy. Values need not be logical.
To me, that means they differ in kind. Scientism is a simple religious position, reflecting values and not facts, and not based on any deduction or observation, but only on our cultural bias. Logical Positivism is an exploration of meaning in a constrained and disciplined way, which ultimately fails, but provides value of its own.