Learn English – The difference between ‘way’ and ‘manner’

In this context, manner and way are identical in meaning. Dictionary references are not particularly helpful, as the relevant meaning of each is defined in terms of the other. For example, the Oxford dictionary defines way as A method, style, or manner of doing something. Manner is perhaps slightly more formal.

They live in a beneficial way through education.

Whichever is used, the resulting sentence is flawed and meaningless. What the writer probably wants to say is:

Their lives are better because they are educated.

The problem is that beneficial is an adjective that describes the source of the benefit, not the recipient. In this sentence, education is the source of the benefit. We can move education to the front, and then through is not required. The recipient (their lives) needs to be attached using a preposition for or to. The sentence can therefore be rewritten as:

Education is beneficial to/for their lives.

The adverbal construct in a xxx way is much more widely used in Arabic (bi-il-taryqah xxx) than in English, as natural adverbs are few and far between in Arabic: English written by arabic speakers therefore tends to contain this construct even when it's not necessary in English. If the writer really wanted to stick to an adverbal form, then the English adverb well would be a perfect choice:

They live well through education.
They live well because of their education.