To find the mean distance between two distinct dots in a 4x4 grid, we need to calculate the total distance for all pairs of dots and then divide by the number of pairs. We will use casework to count the number of pairs for each possible distance.

1. Calculate the number of pairs for each distance:

- Distance 1:
Each dot has 2 adjacent dots (one horizontal and one vertical), except for the dots on the edges and corners.
- Corners (4 dots): Each has 2 adjacent dots.
- Edges (8 dots): Each has 3 adjacent dots.
- Interior (4 dots): Each has 4 adjacent dots.

Total pairs:
$$4 \times 2 + 8 \times 3 + 4 \times 4 = 8 + 24 + 16 = 48$$
However, each pair is counted twice, so we divide by 2:
$$\frac{48}{2} = 24$$

- Distance 2:
- Horizontal or vertical distance of 2:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
- Diagonal distance of 2:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
Total distance 2 pairs:
$$18 + 18 = 36$$

- Distance 3:
- Horizontal or vertical distance of 3:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
- Diagonal distance of 3:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
Total distance 3 pairs:
$$18 + 18 = 36$$

- Distance 4:
- Horizontal or vertical distance of 4:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
- Diagonal distance of 4:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
Total distance 4 pairs:
$$18 + 18 = 36$$

- Distance 5:
- Horizontal or vertical distance of 5:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
- Diagonal distance of 5:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
Total distance 5 pairs:
$$18 + 18 = 36$$

- Distance 6:
- Horizontal or vertical distance of 6:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
- Diagonal distance of 6:
- Corners: Each has 1 pair.
- Edges: Each has 2 pairs.
- Interior: Each has 4 pairs.

Total pairs:
$$4 \times 1 + 8 \times 2 + 4 \times 4 = 4 + 16 + 16 = 36$$
Each pair is counted twice, so:
$$\frac{36}{2} = 18$$
Total distance 6 pairs:
$$18 + 18 = 36$$

2. Calculate the total distance:
$$\text{Total distance} = 1 \times 24 + 2 \times 36 + 3 \times 36 + 4 \times 36 + 5 \times 36 + 6 \times 36$$
$$= 24 + 72 + 108 + 144 + 180 + 216 = 744$$

3. Calculate the number of pairs:
$$\binom{16}{2} = \frac{16 \times 15}{2} = 120$$

4. Calculate the mean distance:
$$\text{Mean distance} = \frac{744}{120} = \frac{62}{10} = \frac{31}{5}$$

5. **Find $m + n$:**
$$m = 31, \quad n = 5$$
$$m + n = 31 + 5 = 36$$

The final answer is $\boxed{36}$