Given two samples that have the same mean, standard deviation, and N: are the values in each sample identical? If not, are there any restrictions that would need to be imposed to ensure that the values of the two samples would be identical? I apologize in advance if this is such a basic question.
 A: 
No. Many data sets can yield the same mean, SD and n.
In the graph above, the three data sets on the left of each graph all share the same mean, SD and n. So do the three data sets on the right of each graph. This is Figure 1 from: Weissgerber, T.L., Milic, N.M., Winham, S.J., and Garovic, V.D. (2015). Beyond bar and line graphs: time for a new data presentation paradigm. PLoS Biol 13: e1002128.
A: 
Given two samples that have the same mean, standard deviation, and N: are the values in each sample identical?

In general, not unless N=2 in both samples. If N is larger than 2, they can differ.
You can see this simply by trying it with some simple cases.
Perhaps the easiest case is to take an asymmetric sample of size $N=3$ and flip it around its mean ($2,3$ and $7$ have a mean of $4$; if you take a new sample of $6,5$ and $1$, respectively, it has the same mean and the same magnitude of deviations from the mean as the original sample, so it will have the same variance and hence standard deviation).
For another example consider these three samples of size three that have the same standard deviation:
Set A: $-1, 0, 1$
Set B: $-a, -a, 2a\quad$  (where $a = \sqrt{\frac13}$)
(i.e. approximately -.57735,-.57735,1.1547)
Set C: $-2b, -b, 3b\quad$   (where $b = \sqrt{\frac17}$)
(i.e. approximately -0.7559289, -0.3779645,  1.1338934)
These all have mean 0 and sd 1. You can make any other mean and sd from these by multiplying by the desired standard deviation and then adding the desired mean.

If not, are there any restrictions that would need to be imposed to ensure that the values of the two samples would be identical?

Sure, you need additional restrictions that reduce the available degrees of freedom for the data down to 0. These restrictions might take many forms, e.g. specifying skewness, or the median or the sample maximum, etc.
Not all additional restrictions will always work to reduce the free dimensions by one (with some sets of existing restrictions, some additional restrictions may be redundant), but that's usually what it takes.
