As others have said, switching an 8 bits/channel file to 16 bits/channel does not gain anything - but neither does it lose anything. Those 256 levels in the original file are just slotted in to 256 distinct levels from the 65,536 available in a 16-bit/channel file. (Actually in Photoshop 16 bit there are 32,769 levels but that doesn't matter here).
However, when you then go on to make adjustments then 16bits/channel does have an advantage. In 8 bits/channel each adjustment calculates a new level and that is rounded into one of the 256 available levels. Then the next adjustment does the same and those rounding errors start to add up and you see stepping in the image (or combing in a histogram). By working in 16 bits/channel, values still are rounded but the number of levels means that the errors are much smaller and it takes an awful lot of cumulative adjustments before any become visible (if they ever become visible at all).
As an analogy. It's like having a ruler and marking points at 1,2,3, 4, 5 6 ....12 inches. Measuring those points with a second ruler accurate to 1/10 of a millimetre, does not make any difference to the position of those points.
However if you then make a series of calculations that will move those points and say each calculation has to be rounded so that the new position falls on a ruler point, using the 1 inch ruler will mean that after each calculation the result has to be slotted against distinct full inches. Whereas with the second ruler, all adjustments can be slotted into 1/10 of a millimetre steps. Which will be more accurate after several, even small, adjustments? At the end, the final result may be slotted back into the 12 inch ruler steps (the equivalent of exporting as an 8 bit file) but that final rounding done once is better than a series of roundings at each calculation.
Dave