Times New Roman is Microsoft's name for the Trueype version of Times New Roman PS, a narrower version of monotypes classic type. The PS version was introduced at the request of Microsoft Monotype in order with the dimensions of the Times Roman (one of the PostScript coincide -Kernschriften Linotype). She has finer capitals that were originally designed for printing German texts (because here, in contrast to the English seem very much more capital letters).
Something from the English Wikipedia page for Times New Roman:
The outcome of all of the legal maneuverings is that Linotype and its licensees like Adobe and Apple continue to use the name "Times Roman", while Monotype and its licensees like Microsoft use the name "Times New Roman".
During the decades of transatlantic "sharing" of the Times designs, and the transfer of the faces from metal to photo to digital, various differences developed between the versions marketed by Linotype and Monotype. Especially these became evident when Adobe released the PostScript version, for various reasons having to do with how Adobe produced the original PostScript implementations of Times. The width metrics were different, as well as various proportions and details.
In the late 1980's, Monotype redrew its Times New Roman to make it fit exactly the proportions and metrics of the Adobe-Linotype version of Times Roman. Monotype claimed that its new version was better than the Adobe-Linotype version, because of smoother curves, better detailing, and generally greater sensitivity to the original designs done for The Times and Monotype by Victor Lardent, who worked under the direction of Stanley Morison. During the same period, Adobe upgraded its version of Times, using digital masters from Linotype, which of course claimed that it had a superior version, so there was a kind of competition to see who had the most refined, sensitive, original, genuine, bona-fide, artistically and typographically correct version. Many, perhaps most, users didn't notice and didn't care about these subtle distinctions, many of which were invisible at 10 pt at 300 dpi (which is an em of 42 pixels, a stem of three pixels, a serif of 1 pixel, and so on).
When Microsoft produced its version of Times New Roman, licensed from Monotype, in TrueType format, and when Apple produced its version of Times Roman, licensed from Linotype, in TrueType format, the subtle competition took on a new aspect, because both Microsoft and Apple expended a great deal of time and effort to make the TrueType versions as good as, or better than, the PostScript version. During the same period, Adobe released ATM along with upgraded versions of its core set of fonts, for improved rasterization on screen. Also, firms like Imagen, now part of QMS, and Sun developed rival font scaling technologies, and labored to make sure that their renderings of Times, licensed from Linotype in both cases, were equal to those of their competitors. Hence, the perceived quality of the Times design became a litmus for the quality of several font formats. Never before, and probably never again, would the precise placement of pixels in the serifs or 's' curves etc. of Times Roman occupy the attention of so many engineers and computer scientists. It was perhaps the supreme era of the Digital Fontologist.
As for the actual visual differences in the designs, well, like any good academic author, I leave the detection and analysis of those "as an exercise for the reader".
Linotype licensed its version to Xerox and IBM, then Adobe and Apple, guaranteeing its importance in digital printing by making it one of the core fonts of the PostScript page description language. Microsoft's version of Times New Roman is licensed from Monotype, hence the original name. For compatibility, Monotype had to subtly redraw their design to match the widths from the Adobe/Linotype version. It has the lighter capitals that were originally developed for printing German (where all nouns begin with a capital letter). Versions of Times New Roman from Monotype exist which vary from the Linotype metrics (i.e. not the same as the version for Microsoft). In addition, the original digitisation of Times New Roman omits automatic ligature insertion, which the version of Times installed with OS X has. (This results in unsightly character collisions with the system version of Times New Roman whenever the characters 'fi' are needed.)