C

Step 1: Analyze the motion of both stones under the influence of gravity. Since we neglect air resistance, the only force acting on both stones after they are thrown is gravity, which accelerates them at a constant rate of $g$ (approximately $9.81 ext{ m/s}^2$).

Step 2: Let's denote the mass of the lighter stone as $m$ and the mass of the heavier stone as $2m$. The initial speed for both stones is $u$.

Step 3: For the stone thrown upward (lighter stone), it will rise before it comes to a stop and then fall back down. The total time taken to reach the maximum height and subsequently the time taken to fall back down to the ground can be determined using kinematic equations. The stone will spend more time in the air compared to the stone thrown downward.

Step 4: For the stone thrown downward (heavier stone), it will continue to accelerate downwards from the moment of release.

Step 5: To find the final velocity of both stones when they hit the ground, we can use the equation:
$$v^2 = u^2 + 2as$$
where $v$ is the final velocity, $u$ is the initial velocity, $a$ is the acceleration (which is $g$), and $s$ is the height of the tower.

Step 6: Since both stones are thrown with the same speed $u$ and fall from the same height $s$ under the same acceleration due to gravity, the final velocity just before hitting the ground, for both stones, can be calculated by substituting their respective values into the equation.

Final Calculation: Both stones will have:
$$v^2 = u^2 + 2gs$$
Thus, both will attain the same final velocity upon hitting the ground if they are thrown from the same height, despite being of different masses. Therefore, they both hit the ground with the same speed.

Thus, the correct answer is: Option C: Both the stones will have the same speed when they hit the ground.