“我把爸爸的生日忘了”强调“忘了”这种结果。 “我忘了爸爸的生日”更强调“忘了”这一种行为

这是同一个意思的两种表达方法。简单说，意思完全一样。

Yes, they are pretty much the same. “把 + object + verb” = “verb + object” Maybe there is a very slight ambiguity. “我把爸爸的生日忘了。” gives me a feeling like maybe you forget to celebrate your father’s birthday. “我忘了爸爸的生日” sounds like you forget what date is his birthday. =“我忘了爸爸的生日(是几号)” "把" put a stress on the object

是的，没有什么区别，“把”可能更加重了语气。类似于英语“I forgot my father‘s birthday”和“I had my father‘s birthday forgotten” , ofc they are exactly the same and comparable, but I think you get the idea. 我把他的花瓶打碎了( I had his vase broken)