Neither of those is an idiom or a stock phrase.
Therefore we would interpret them just according to the meanings of their words, and they are not the same.
"As the case stands" may well refer to a specific legal matter. If not, it is probably a variant of the oft-used phrase "As matters stand". Either way, it introduces a remark about a mix of facts, circumstances, and possibly arguments. In contrast, the other one is just about a single line of argument or reasoning.