月末ごろ and 末日: around month-end and the last calendar day
Compare an arrival around month-end with a check on the last day of April, also reading 月末 in deadline wording.
Around the end of a month
月末, read げつまつ, means the end of a month. 月末ごろに荷物が届きます。 says that the luggage will arrive around month-end.
ごろ means that one exact date is not specified. Ask 何日に届きますか。 when the arrival date is needed.
The final calendar date
末日, read まつじつ, means the last day. 四月末日に確認します。 schedules a check on the last day of April, April 30.
二月の末日は年によって違います。 says that February’s last day differs by year. Identify the year before finding the final calendar date.
月末 can also identify the last day
月末までに提出してください。 requests submission by month-end. In deadline wording, 月末 can identify the last day rather than only a vague period.
This comparison distinguishes the approximate timing in 月末ごろ from the specific date in 末日. Do not treat 月末 and 末日 as meanings that can never overlap.
Does 月末ごろに荷物が届きます。 supply an exact arrival date?
Show the answer and why
No. It gives an arrival around month-end. A specific date requires additional information.