Is there any statistical difference between 10 Bernoulli trials and 1 binomial trial with parameter n = 10? By "statistical difference" I mean literally any difference between the two of statistical import, beyond of course the notation or words used.
 A: Yes they're different in a simple, fairly obvious sense.
Ten Bernoulli trials (presumed to be independent with with common parameter $p$) is ten 0/1 values, so an observation on it looks like "$(0,0,0,1,0,1,1,1,0,0)$".
The sum of such a vector is distributed as $\text{binomial}(10,p)$; an observation on that looks like "$4$".
With an observation from the first thing (ten Bernoulli trials) I can answer a question like "was there a 1 on the third trial?" or "what's the difference in the number of 1's in the first 5 trials and the second 5 trials?". With the second thing (one observation on a binomial), I simply cannot consider those questions, since I have compressed the information from ten values to a single number. The first thing contains more information (in a particular sense) $-$ but if the assumptions are true it's not got any more information about $p$; the sum is sufficient for that.
For example, if you want to answer a question "is my binomial model reasonable?" you might want to consider the possibility that $p$ might change over the course of the trials, or that the trials may be serially correlated. You can't begin to assess that from one binomial value (you've thrown out the information that would let you discern if the binomial model was suitable) but you can consider such a question from a set of $n$ Bernoulli trials.
Therefore there's a clear "difference of statistical import" here.
[Ten trials is small enough that you can't get much power in relation to those questions, but with a binomial you simply can't address it at all. If I had say a hundred trials, though, or a thousand, I could do some more interesting things to address those questions, and might hope to identify the extent to which my model was not really describing the situation.]
