Top down approach doesn't visit all the possible states i.e it only visits the required states .
In bottom up since we are filling the table we have to visit all the states.
So technically top down is more efficient than bottom up but there are other factors like recursion overhead for top down method .
Also top down is easy to code compared to bottom up.
Consider the case for finding binomial coefficient using this formula
nCr = n-1Cr + n-1Cr-1
Now use TOP DOWN method for finding 4C3
4C3 = 3C3 + 3C2
3C2 = 2C2 + 2C1
We stop beacuse 3C3 ,2C2 ,2C1 are all trivial cases .
Now using BOTTOM UP method you would do something like
For I = 1 to I = 4
For j = 1 to j = I
//Use bottom up Using this table filling
Here you fill values like 3C1 ,4C1,4C2 .
Notice that this values won't be calculated in top down method specified above .
Hope it helps .