Hi, Nics. I reckon the answer to your question is simple enough. 别 and 不要 are of the same meaning, so you can treat them as equals. However, when the question comes to how to distinguish them, I have to admit, even if I am a native Beijing guy, I can hardly tell any difference. If there is any, it may be just, in the case of 'don't forget', 別 is used more frequently than 不要 in Beijing dialect as well as oral Madarin. In the real world, we are more likely to say 别忘了 than 别忘记 and 不要忘记, though they imply the same meaning. It is maybe because 别 is merely a single character so that it is easier to say out. Morever, it is noted that 忘记 is more of a written word, so we often use 了 instead of 记, which makes this expression more authentic.

strictly speaking, there is no difference between 别 and 不要 in Mandarin 别 used more in oral Mandarin, 不要 sounds like more formal and used in written Mandarin. for example: 别哭＝不要哭 别去＝不要去

usually they are no differences.but some occasions they are different.for example,A:do you want to have a coffee.if B don't want to have it.he should say:不要.not say 別.And 別 is Colloquial,is not used in formal occasions.And 別stand for "please don't do what",不要 stand for "don't do what".my english is poor,can you understand?

they are the same meaning but we usually use别忘记，that is a native expression.

不要忘记 is more normal than 别忘记，and in the spoken language ,we usually use 别忘记。However，不要忘记 emphasize the importance of something.