To add some numbers: Assume that our deflection takes place 5 years before the asteroid would hit Earth centrally.
1) If you just want to avoid the collision, rhe required velocity change has to change its path by about 10000km after 5 years, or 0.06m/s. The asteroid would come very close to earth, but miss it. The final orbit would still cross the orbit of earth, so the object could become dangerous at some point in the future (probably millenia or more) again.
2) If you want to change the orbit so much that it does not even come close to earth, you have to move it by ~1 million km in 5 years, or 6m/s.
3) If you want to shoot it out of our solar system, you have to change its velocity by roughly 10000m/s.
4) If you want to drop it into sun, you have to change its velocity by roughly 30000m/s.
While there are cheaper ways to do (3) and (4), they require much more time and very good planning. (1) and (2) are way easier. If we just have 1 year in advance, the numbers for (1) and (2) have to be multiplied by 5, but they are still orders of magnitude below the others.