What is the difference between phrases "by the afternoon" and " by afternoon"? In some student books said that you should use only "by the afternoon" but I met many times the phrase"by afternoon". So are both phrases acceptable and do they have the same meaning?

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In the United States, they are both common, correct, acceptable, and mean the same thing.

There might be some subtle difference in a shade of meaning, but I'd just be guessing if I tried to say what it is.

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They're both acceptable. There are often cases with 'time periods' where the article can be omitted or included; such as, periods of the year (i.e. seasons: spring, summer fall, etc) and periods of the day (i.e. morning, afternoon, evening).

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Personally I consider both ‘by afternoon’ and ‘by the afternoon’ a little vague compared to ‘morning’ and ‘evening’ and wouldn’t say either. Maybe I’d say ‘by early/late afternoon’