If you can use one of the words "will" or "would" in a sentence, then you can almost always use the other. The meaning, however, does change. Learning the subtle differences between modal verbs is one of the most important (and difficult) tasks that someone who wants to learn English needs to master.
The primary role of the word "will" in English is to express conviction or certainty. Although it is often described as the key component of a "future tense", that importance is greatly overstated. In
"he promised he will come tomorrow"
"will" is used to express the firmness of a promise. It is not needed for "future" because the word "tomorrow" by itself does that very well:
"he promised he is coming tomorrow".
If you drop "tomorrow", then "will" can help to put the promise into the future:
"He promised he will come".
If you use "would" instead of "will", the effect is to lessen conviction and certainty, thereby introducing doubt or conditionality. "Would" also softens the sentence, making it less forceful. That is why "would" always seems more polite when you make a request: "Would you please pass the salt?" The sentence
"he promised he would come tomorrow"
is less precise than the one using "will". It could mean many different things. It could imply "he didn't come and broke his promise". It could mean "he stated he is coming tomorrow". It could mean "he might come tomorrow. At least, he promised". It could mean many things. Only context can provide an exact meaning. It is easier to say what it does NOT mean than what it does mean. It does not mean that he promised he will certainly come tomorrow.