Compound interest problem using PMT in Excel
The reason of this difference is that you do not fully understand what Compound interest is.

Actually, if you want to get the same amount as the site, while using your monthly payment, which is almost correct (more about this "almost" thereafter), what you must do is less straightforward than simply multiplying it by $120$ (calculation which would return the correct value in the $0$ interest rate case.)

What you must understand is that, the first monthly payment, let call it $P_0$, is going to capitalize (at a $\frac{5}{12}\%$ rate) for $120$ months, $P_1$ during $119$ months, $...$ and $P_{119}$ during $1$ month. And this is the sum of all these differently capitalized identical payments that you must compute to get $8235.05$. 

That being said, if you do so, you will get $8263.36$ instead of $8235.05$, since the payment is actually due at the beginning of the period (whence the term almost above), i.e. you must set a $1$ instead of $0$ for the argument Type, which will return a monthly flow of $(\$52.81)$. 

Thus, to summarize

$8235.05 = 52.8127046833568 \times \sum_{i=0}^{119} \left( 1+\frac{.05}{12} \right)^{120-i} \approx 52.813 \times 155.9292889
$

Where $155.9292889$ can be interpreted as an implicit capitalization factor.