I am wondering whether legally a single patent with Claim A and Claim B allows the same legal protection for the content presented in Claim A and Claim B as two single patents containing in Patent 1 claim A and patent 2 Claim B?
There are multiple ways to end up with two patents with claims that cover related subject matter. One is by filing two applications and the other is by having a continuation or divisional from a single parent or some more complicated filing relationship using more than one continuation or divisional. In the case of two applications they might or might not be filed on the same day.
If filed as two applications there might or might not be an issue of double patenting depending on the exact subject claimed. The infringing product might infringe two claims neither of which is obvious in light of the other. This might be the case with two claims in one patent or in the case of two patents with one claims each. As mentioned in another answer there may be an issue of double patenting that was not caught by the examiner - if so a terminal disclaimer may be filed to obviate that issue.
If two claims in an original parent application are related but the examiner makes a restriction requirement (the claims are in different classes and it would be a burden to search both, for example) and the applicant does not traverse, then one claim can be pursued in the parent and the other claim can be refiled in a divisional. By the safe harbor (U.S.C. § 121) the claim in the divisional is by definition not subject to a double patenting problem. You end up with two patents of one claim each with zero double patenting issue.
The two cases have the same legal effect on the face of it. It is possible that scrutiny of the history of a patent prosecution may turn up some irregularity that can get a patent torn up. Two patents from two applications that have no formal relationship to each other might present a more difficulty target for invalidation. If the infringer was to use a USPTO process (IPR) to try to invalidate a patent it will cost more to try to invalidate two patents.
This answer is with respect to US law and practice at the USPTO.
If claims A and B are obvious variations of one another, they should be part of the same patent. When such claims are submitted in separate patent applications, the later-filed application should be rejected on the basis of double patenting. Double patenting is the idea that the same invention, or obvious variations, should not be patented twice, because the patent term should not be extended by the filing of sequential patents and because the ownership of the same invention by two different entities would cause undue conflicts. The Applicant can respond by filing a terminal disclaimer in both applications. A terminal disclaimer is a statement by the patent owner to disclaim any time period of the second patent that extends beyond the expiration of the first patent. Also, the patents A and B must maintain common ownership.
The patent owner pays maintenance fees on both patents subject to a terminal disclaimer.
If claim A and claim B are in separate patents and are not subject to a terminal disclaimer, then on the face one assumes that they are separate and distinct inventions. They would provide different coverage for the different inventions. If they are found to be obvious variants, then the earlier-filed claims can invalidate (make obvious) the later-filed claims.