For pure self-similar fractals, they are the same by Moran Theorem (or see Section 9 in Falconer's book).

For other sets they can be really different. For example, if A=the rational in [0,1], the Hausdorff dimensino is 0 while the Box-Conting one is 1.

Thank you very much Miguel Sánchez-Granero. As you said, according Moran theorem both the dimensions are same. But, for other sets (for example if A = the rational in [0,1]), why are these dimensions different and how? For random fractals like fractal terrain how is the difference between two dimensions?

Iif A = the rational in [0,1], the Hausdorff dimension is 0, since it is countable, but the box-counting dimension is 1, since you will have the same number of boxes that you have for the interval [0,1].

For random fractals like fractal terrain, I do not know, maybe most of the times they are the same or sometimes they are not. Anyway, how do you calculate the Hausdorff dimension for such sets?

in general Hausdorff dimension is used in theory for mathematical purpose, and the box counting is used in pratical case, because it's easy to implement

Hausdorff dimension (s) : In calculating s ,we assign different weights , |Ai|s to the covering sets Ai ; whereas 

In box counting dimension  :we use same weights ks  for each covering set.It indicates the efficiency with which a set may be covered by small sets of equal sets , whereas Hausdorff dimension involves covering by sets of small but perhaps varying size

Dear Arvind, you are right. F M Barnsley's 'contraction mapping theorem' is also useful to clarify this difference. In his method, your Ai are the contractivity factors and they can have varying magnitudes.  While, in box counting dimension this contractivity factor is always same.

الملف المرفق يوضح الفرق بين ايجاد البعد الهاوزدورفي والبعد الكسوري بطريقة عد الصناديق

See this book.

You can see also this book.

one is finer than other

The Hausdorff dimension is a finely graded tool to measuring the global "thickness" of a metric space. It minimises the scaling exponent of countable coverings of *any* size, whereas the box-counting dimension looks at the scaling of coverings of the *same* size. This is why the Hausdorff dimension of countable sets is 0:

Let (X,d) be countable, then {B(x_n, \epsilon^n) : x_n \in X} is a cover for any small epsilon. If you want to cover a countable set that is dense in, say, [0,1]^d but restrict yourself to coverings of the same size, you will need to (essentially) cover the entire space! You cannot "cheat the system" by a family of quickly shrinking balls.

For self-similar (or self-conformal) sets F, both notions agree because of a global "regularity": Every ball of size r < diam (F) that is centred in F set contains a not "too distorted" copy of r * F (i.e. a not too distorted copy of F scaled down by r). This is the "implicit theorem" in Falconers book.

For random self-similar sets (at least with most reasonable definitions of random) the same is true: the Hausdorff and box-counting dimensions agree because of some rough global "regularity": Some parts can be "thicker" than others, but as the Hausdorff and box-counting dimensions are global (they consider the entire set) these thicker regions get lost in this "averaging" over the whole space.

I may also interest you in the concept of "intermediate dimensions" that try to interpolate between these notions, see e.g. Preprint Intermediate dimensions