Q:

Comparison of the effects of collisions from an NFL Nose Tackle and a Car with roughly the same momenta

If you get hit an NFL Defensive Tackle who runs at roughly 17mph (7.6m/s) it'd hurt a lot, but if you got hit by a normal car at 1.3mph (about 0.6m/s) it hardly hurts at all, and a collision from an extremely slow moving Oil Tanker with a huge mass would likely be almost unnoticeable, but they all have roughly the same momentum, when Force = dP/dt. Why?

A:

Due to the amount of momentum (and energy) transferred to you. Assuming a completely inelastic process (and an idealized one dimensional setup) you get from the conservation of momentum ($v$ is the initial velocty, $p$ the initial momentum, $v'$ is the final velocity, $M$ is the mass of the object hitting you, $m$ is your mass):
$$ p = M v = (M + m) v' $$
$$ v' = \frac{M}{M + m}v$$
If you keep the momentum constant, this means, that $v$ scales as $p/M$, so a tanker with the same momentum will be very slow.
The energy dissipated in the collision process will be:
$$ E = \frac 1 2 Mv^2 - \frac 1 2 (M + m) v'^2 = \frac 1 2 \left(M - \frac{M^2}{M + m}\right) v^2 = \frac 1 2 \frac{mM}{M + m} v^2.$$
Formulated in terms of the momentum we get:
$$ E = \frac 1 2 \frac{mM}{m + M} \frac{p^2}{M^2} = \frac 1 2 p^2 \frac{m}{M (m + M)}. $$
Calculating numbers for the specific cases ($m \approx 100\,\mathrm{kg}$, $M_\text{NFL} \approx 100\,\mathrm{kg}$, $M_\text{car} \approx 1000\,\mathrm{kg}$, $M_\text{tanker} \approx \infty$, $v_\text{NFL} \approx 10\,\mathrm{m/s}$, $p \approx 1000\,\mathrm{kg\cdot m/s}$):
$$E_\text{NFL} = 2500\,\mathrm{J}$$
$$E_\text{car} = 45\,\mathrm{J}$$
$$E_\text{tanker} \approx 0\,\mathrm{J}.$$
While pain and damage done by an object depends on many factors, the most important is probably energy dissipated in your body (more specifically weighted by the volume of tissue). The dissipation volume for the tanker or the car will be pretty large (perhaps half of your body).
The NFL tackle will be done with a rigid point of the tackler's body (something like the elbow?), so the energy will be dissipated in a quite small volume of your body, while the dissipated energy will be much larger than in the car or tanker case. This way you can also understand why being shot does so much damage (although it has even less mometum. Depending on the fire arm, it is on the order of the momentum of a pedestrian). To stop the bullet the amount of energy will be disspated in a very small volume of your body. (As you are much heavier than the bullet nearly all the bullets energy will be lost, and this energy scales like $\propto mv^2$).

