A Latin adjective can sometimes be read either as a mere attribute or more broadly.
For example, consider these two translations:
Homo conscius intelligit.
1. A conscious man understands.
2. A man, being conscious, understands.
In the first translation conscius is a mere attribute, describing what kind of a man is in question.
In the second one there is more than a description, it gives a reason.
If you want, you can read the first one as an attribute and the second one as an apposition.
The second translation does resemble absolute ablatives in a way, but one cannot construct ablativus absolutus with a single ablative adjective.
In this Spinoza passage the positioning of conscius does make me think of the second kind of interpretation.
In prose one would expect an attribute to be somewhat close to the referent.
The adjective appears to have a place in the flow of the argument, which suggests that it has more weight than an attribute.
(I am not familiar with Spinoza's style, so someone more used to him can judge better whether this judgement is sound.)
It would also be possible to read the passage in the first way, which gives something like:
On the other hand, the conscious wise man, insofar as he is considered as such, suﬀers scarcely any disturbance of spirit, but, by virtue of a certain eternal necessity, of himself, of God and of things, never ceases to be, but always possesses true spiritual contentment
This is a possible interpretation, but I think White's choice makes more sense.
To me this a question of emphasis on conscius, and my variation does not have enough of it to my taste in comparison to the original.