"You have my word" is basically the same as "I promise".
However if you tell someone "You have my word on that.", you have made a statement that is more dramatic and carries more weight, emotionally.
If I tell someone that somebody has broken a promise, his first reaction might be to ask what the promise was. The final judgment might depend on what was promised.
However, if I say that somebody has broken their word, the first reaction would be to pass judgement on the that person's honor. The first reaction would be much stronger and more final.
While this is not true in all cases, there would definitely be a tendency to put more weight on someone's word than their promise.