Am I allowed to share the final-version PDF file with him/her (by e-mail)?
From a (US) legal perspective, that depends entirely on the publishing agreement. Some journals do allow this; if so, it will be stated explicitly in the agreement. For example, the APS copyright transfer agreement includes the text
The author(s)...shall have the following rights (the “Author Rights”):
3. The right...to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.
Without something like this in the agreement, whether you're allowed to share the paper depends on whether it can legally be considered fair use. No copyright agreement can prohibit something which falls under fair use, but on the other hand you don't really know what falls under fair use until a court tells you. I'm not sure if there's any precedent on whether sharing copies of an article for private research use qualifies as fair.
I won't address this from a moral perspective since you asked whether it's allowed, but as some of the other answers show, it's possible to reach a different conclusion that way.
I mean, does it make any difference with sharing it publicly (which I do know I am not allowed to)?
Well, this also depends on the publishing agreement. By default, if the agreement doesn't say anything otherwise, there is no difference between sharing the article publicly and sharing it privately; both qualify as unauthorized distribution, and both are similarly illegal. But the publishing agreement may allow for some forms of public sharing. Going back to the APS agreement, it includes several clauses allowing various forms of public sharing (under "Authors' Rights"):
2. The nonexclusive right, after publication by APS, to give permission to third parties to republish print versions of the Article or a translation thereof, or excerpts therefrom, without obtaining permission from APS, provided the APS-prepared version is not used for this purpose, the Article is not republished in another journal, and the third party does not charge a fee. If the APS version is used, or the third party republishes in a publication or product charging a fee for use, permission from APS must be obtained.
3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.
4. The right to post and update the Article on free-access e-print servers as long as files prepared and/or formatted by APS or its vendors are not used for that purpose. Any such posting made or updated after acceptance of the Article for publication shall include a link to the online abstract in the APS journal or to the entry page of the journal. If the author wishes the APS-prepared version to be used for an online posting other than on the author(s)’ or employer’s website, APS permission is required; if permission is granted, APS will provide the Article as it was published in the journal, and use will be subject to APS terms and conditions.
So under this agreement, you (the author) can post the final journal version of the article on your own personal website, and you can post preprints (i.e. versions you prepared prior to submitting to the journal) on a site like arXiv or grant permission to third parties to distribute these versions.