Proper Grammar and Word Choice

They have slightly different shades of meaning. Both are grammatically correct. The first one 'early' implies the first half of the week. If a week starts on Sunday, then we're talking about Sunday through maybe Wednesday at the latest. If the week starts on Monday, then we are looking about Monday through, possibly Thursday on the outside. But we're pushing it with Wednesday and Thursday, which most people would see as the middle of the week, whether that week starts on Sunday or Monday.

For the second one 'beginning' implies a hair earlier than the former. The beginning is much more clearly defined, so it's more likely going to be Sunday or Monday or whichever day is considered to be the first day in the week. At a workplace, that's likely to be further refined to mean the first work day (often Monday, but holidays and shifts can interfere with that).

The truth is, most native English speakers would use the two terms more or less interchangeably. And I'll admit I'm being pedantic. I would accept them as virtually synonymous sentences, too.