To solve this problem, we will start by using the properties of an arithmetic progression (AP).

The sum of the first $n$ terms of an AP can be calculated using the formula: $$ S_n = \frac{n}{2} (2a + (n-1)d) $$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, and $d$ is the common difference between the terms.

Given the information: $$ S_{10} = 390 $$

We can plug $n=10$ into the sum formula to get:

$$ S_{10} = \frac{10}{2} (2a + (10-1)d) $$

$$ 390 = 5(2a + 9d) $$

$$ 390 = 10a + 45d $$

$$ 78 = 2a + 9d \quad .........\text{(1)} $$

Next, we're given the ratio of the tenth term ($T_{10}$) to the fifth term ($T_5$): $$ \frac{T_{10}}{T_5} = \frac{15}{7} $$

The $n$th term of an AP is given by:

$$ T_n = a + (n-1)d $$

So, for the tenth term: $$ T_{10} = a + (10-1)d = a + 9d $$

And for the fifth term:

$$ T_5 = a + (5-1)d = a + 4d $$

Now we can write the ratio as:

$$ \frac{a + 9d}{a + 4d} = \frac{15}{7} $$

$$ 7(a + 9d) = 15(a + 4d) $$

$$ 7a + 63d = 15a + 60d $$

$$ 63d - 60d = 15a - 7a $$

$$ 3d = 8a \quad .........\text{(2)} $$

Now we have two equations (1) and (2):

$$ 78 = 2a + 9d \quad \text{(1)} $$

$$ 3d = 8a \quad \text{(2)} $$

We can solve these equations simultaneously.

From equation (2):

$$ d = \frac{8}{3}a $$

Plugging this back into (1):

$$ 78 = 2a + 9\left(\frac{8}{3}a\right) $$

$$ 78 = 2a + 24a $$

$$ 78 = 26a $$

$$ a = 3 $$

Now we can find $d$:

$$ d = \frac{8}{3}a $$

$$ d = \frac{8}{3} \times 3 $$

$$ d = 8 $$

Now we can find $S_{15}$ and $S_5$ using the formula for the sum of an AP.

For $S_{15}$:

$$ S_{15} = \frac{15}{2} (2 \cdot 3 + (15-1) \cdot 8) $$

$$ S_{15} = \frac{15}{2} (6 + 14 \cdot 8) $$

$$ S_{15} = \frac{15}{2} (6 + 112) $$

$$ S_{15} = \frac{15}{2} \cdot 118 $$

$$ S_{15} = 15 \cdot 59 $$

$$ S_{15} = 885 $$

For $S_5$:

$$ S_5 = \frac{5}{2} (2 \cdot 3 + (5-1) \cdot 8) $$

$$ S_5 = \frac{5}{2} (6 + 4 \cdot 8) $$

$$ S_5 = \frac{5}{2} (6 + 32) $$

$$ S_5 = \frac{5}{2} \cdot 38 $$

$$ S_5 = 5 \cdot 19 $$

$$ S_5 = 95 $$

The difference $S_{15} - S_{5}$ is:

$$ S_{15} - S_{5} = 885 - 95 $$

$$ S_{15} - S_{5} = 790 $$

Therefore, the correct answer is Option C, which is 790.