The number of bits tells how many values are possible per color component. This specifies bit-depth as BPC (bits-per-component) which is what Photoshop uses. Windows on the other hand uses BPP (bits-per-pixel) which is why you will see 24-bit colors which is the same since there are 3 color-components: Red, Green and Blue.
An 8bit file therefore allows 256 different levels on the Red axis, 256 on the Green and 256 on the blue axis, since 2 to the power of 8 is 256. So, when you multiply these out you get 16,77.216 possible colors within the color-space of the file (Most times sRGB or AdobeRGB).
A 16-bit file works the same way except that each component can have one of 65,536 values. So, each pixels get 48-bits which gives 28,147,497,6710,656 (roughly 28 trillion) possible values within the color-space. While this is much more precise, it does not allow you to represent colors outside of the color-space, only more variations within.
When you paste an 8-bit image into a 16-bit image nothing immediately changes because your source image is 8-bits so it only specifies colors among 16 millions of the possible 28 trillion colors in a 16-bit file. This is why you do not see any difference. Imagine you have a 16 megapixels 8-bit image with all possible 16 million colors. Should you paste it into a 16-bit file, the 16-bit file will only use 16M of its possible 28T colors.
Now if you were to start manipulating that image, distorting, applying filters, etc, the result would be slightly different than had you applied the same changes to an 8-bit image since the calculations to compute the effects of manipulation would be done with greater precision.
While the precision is twice is much, you really start seeing the difference after a lot of manipulation. Your screen is actually hiding most of the difference since the vast majority of displays only support 8-bit color, some do 10-bits but that is it. All the extra bits are there to improve calculations but they are no readily visible due to limitations of computer screens.