Are the literal translations of the two phrases the same?
If they have the same meaning, why would people use '的' in a phrase like this? Would it make a phrase more polite or beautiful?
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If translated literally, your two phrases would be fly freely and freedom's flying. In this case (the way I'm interpreting it), the 的 functions as a bit of a possessive. If you meant 地, which in this case would turn the 自由 into an adverb (freely instead of free or freedom), then there isn't much of a difference in meaning. However, the four letter phrase is a common trend in Chinese and is often associated with more poetic and metaphorical statements, so would be more beautiful, yes.
(If you are confused about the finer details of the 的、得、地 grammatical construct, in general 得 is used between adverb and verb when the verb is first, and 地 in the reverse situation, while 的 is usually between adjective and noun.)
both are ok. in real practice, I will read them slightly different. I may put a very short pause in the middle of 自由-翱翔, whereas I will read it directly through for 自由的翱翔. but that really doesn't matter in terms of its meaning and other effects (beautiful or polite). what will make it more attractive is the way you read it.
They are literally same.
The difference is subtle, 的 splits the adjective and noun, make the adjective/ noun stand out from the whole phrase.
Anytime you want to add more weight to the adjective/ noun , add 的. If you want to be more concise or rhythmically with other phrase , omit 的.
Omit 的 is better, since the first phrase is 4 characters.
Here with 的, you can emphasis 自由 or 翱翔. And If you want to omit 的, you can, it's a personal choice.
This answer explains in an East Asian way
It is a common usage in East Asia, "topic indicator", you can call it. Although it can vary a lot structurally, (the specific usages may confuse you, ) the purpose are actually quite similar, to make the structure clearer and so that you can stress some part of your sentence. And to some extent, Because tones are used for distinguish meanings, a few "indicator" word are special replacement for these "**tone-based **" structure.
Anyways, (if you prefer a short version), the difference here(with or without 的) are usually expressed in different tones in English.