"I promise" is used in personal conversations. "You have my word" could be used more in business. You would never say "I promise" in a business setting or meeting. Cheers...

"I promise" is casual and informal. "You have my word" is more formal, more serious. You might use this phrase if someone doubts your promise. This phrase is also a little more old-fashioned and not used very much in everyday conversation. More commonly found in literature and less in spoken English.