What exactly is the difference between $3 \mathrm m/\mathrm s^2$ and $3 \mathrm m/\mathrm s$? According to Wikipedia...
An object experiences a constant acceleration of one metre per second squared (1 m/s²) from a state of rest, when it achieves the speed of 5 m/s after 5 seconds and 10 m/s after 10 seconds.
From this, it seems like that $1 \mathrm m/\mathrm s^2 = 1 \mathrm m/\mathrm s$, though that wouldn't make sense considering how they're treated as separate units.
On
The first one is an acceleration, whereas the second one is a velocity.
For cars it would be "from 0 to 100mph in 3 seconds" (or, strictly speaking, the inverse of that) vs. maximum speed "200mph"...
On
The first indicates an acceleration, the second is a velocity.
The acceleration value indicates that with each passing second the associated velocity increases by 3m/s.
While the second value idicates a velocity that does not change with respect to time. As each second passes, the velocity remains at 3m/s.
Consider a snapshot of a space rocket starting with its engines burning, exactly at the moment when it begins to move. Its velocity is still $0 \frac{\mathrm{m}}{\mathrm{s}}$, but its acceleration can be over $20 \frac{\mathrm{m}}{\mathrm{s}^2}$.
Now, imagine a shuttle in space. It might have huge velocity, but without its engines burning (and disregarding the gravity of the Sun, the Earth, the Moon, etc.), its acceleration is zero.
I hope this helps $\ddot\smile$