Okay then, I'll base my own answer simply on an analysis of the currently accepted answer.
The key sentence in the accepted answer (I admit that you are right suggests that the speaker wasn't agreeing with the second person's opinion but now admits or accepts openly that the second person is right) is simply wrong.
I will concede that 'admit' is admittedly the basis of the definition of 'concede'. But I will not admit that 'admit' itself necessarily implies the existence of a prior dispute. 'Concede' does carry that implication, but the difference between them is that 'admit' does not.
If you appear in a criminal court, you will be asked, firstly, to enter a plea: to plead guilty or not guilty, to state whether or not you admit the charge. The court will ask this on your first appearance before it, even though you have not previously had an opportunity to deny the charge. At this first hearing you can only either admit or deny it. If you deny it, then at a subsequent hearing you might change your mind and opt to concede defeat by changing your plea.
'Admit' does imply some consideration has taken place before speaking out, but it does not carry an additional implication that the speaker has disputed the facts prior to speaking. To maintain the fictional court setting: it is possible to admit the charge at the first hearing, by pleading guilty, an admission necessarily not requiring that you had disputed your guilt previously, since the charge had never previously been put to you.
There is nothing to concede, if you initially do plead guilty, since concede does carry an implication: namely that the speaker had previously made a false - or, at least, an inconsistent - statement (had, for instance, previously entered a plea of 'not guilty', knowing it was not true).
'Concede' implies a change of mind. 'Admit' only implies a making-up of one's mind. The distinction is between making a decision and changing a decision once made.
Best Answer
Merriam–Webster's Dictionary of English Usage (MWDEU) has an entry for 'admit of'. It quotes Fowler (1926) pointing out "that the combination 'admit of' is more limited in application than it once was and that it usually takes a nonhuman subject" (emphasis mine).
MWDEU lists the following examples of use with a nonhuman subject:
and the following example of the rare usage with a personal subject:
Hence, to answer the OP's question, both sentences:
have essentially the same meaning. Although the latter version is rarer, it has, as Fowler points out, a nonhuman subject.