Why would someone use #define to define constants?

It's simple question but why would someone use #define to define constants?

What's the difference between

#define sum 1 and const int sum = 1;

#define has many different applications, but your question seems to be about one specific application: defining named constants.

In C++ there's rarely a reason to use #define to define named constants.

#define is normally widely used in C code, since C language is significantly different from C++ when it comes to defining constants. In short, const int objects are not constants in C, which means that in C the primary way to define a true constant is to use #define. (Also, for int constants one can use enums).

No one should not!
Actually, One should prefer const int sum = 1; over #define sum 1 for a number of reasons:

Scope Based Mechanism:

#defines don't respect scopes so there is no way to create a class scoped namespace. While const variables can be scoped in classes.

Avoiding Weird magical numbers during compilation errors:

If you are using #define those are replaced by the pre-processor at time of precompilation So if you receive an error during compilation, it will be confusing because the error message wont refer the macro name but the value and it will appear a sudden value, and one would waste lot of time tracking it down in code.

Ease of Debugging:

Also for same reasons mentioned in #2, while debugging #define would provide no help really.

Thus, to avoid the above situations const will be a better choice.

For the example that you just gave, I would normally use a const. Except of course, the #define can be used for conditional compilation elsewhere:

#if SOME_DEFINE == 1
    // Conditional code
#endif

This is something you can't do with a const. If you don't need the value to be accessible from the preprocessor, I'd say use a const unless there's some reason why that's not possible. There's some stuff on this in the C++ FAQ lite, where they rightly point out that just because the preprocessor is "evil", it doesn't mean you'll never need it.