So is there any differences between 忘 and 忘记？They both mean to forget but do they have any other differences in terms of usage, location within a sentence, etc
Please read Result complement
Also 了 : 'aspect marker' indicating completed action
'忘' (forget) as a verb cannot be used without a result complement or aspect marker
掉 is the result complement of 忘.
It describes the result of forget is the object is 'forgot away' (from one's mind)
You cannot remove the result complement '掉' from '(忘掉)自己的手電號碼' and write '(忘)自己的手電號碼'
了 is a aspect marker that indicates 'completion of an action'
You cannot remove the aspect marker '了' from '(忘了)自己的手電號碼' and write '(忘)自己的手電號碼'
'忘' = 'forget'
'记' = 'remember/ recollect'
'忘记' (forget to remember) = 'to forget'
'忘记' is a compound word verb
It can be used with or without a result complement or aspect marker
忘记自己的手電號碼 = forget one's own cell-phone number
忘记(掉)自己的手電號碼 = forget one's own cell-phone number (away)
(忘记了)自己的手電號碼 = (forgotten) one's own cell-phone number
Terms like '忘本'，'忘情' that directly apply the verb 忘 before a noun, are in effect, compound words as in "做人忘本" (being ungrateful), 忘情演出 (passionately perform).
The other cases of 忘 directly apply before a noun are 1. 'in idiomatic form', like "見利忘義" 2. 'with an adverb', like 難忘, 不忘, 未忘, and they are all literary terms
see dictionaries (esp. examples): 忘记 forget disremember 忘记;忽略 lose sight of 忘:忘却。忘怀。忘我。忘情。忘乎所以, showing 忘 only occurs in a limited number of fixed (time-honored) expressions, including 成语。 忘记 explained: 合成词 of type B （see e.g. 实用汉语语法，词的构成，３。２。合成词的构造方式 ３。２。１。联合式合成词。。。B：两个语素的原义发生了变化：有的产生了新义，有的只保留部分原义。如＂东西、开关、矛盾、买卖、忘记＂。Also see baike.baidu,联合型（并列型）：举例：3. 两词根意结合后，只有一个词根的意项在起主要作用。 例如：国家 人物 忘记 没有 质量