Hey there itsmitry,
Thanks for reaching out to the Apple Support Communities.
To get started, while we understand you mainly use Gmail with the Mail app, do you have any other email accounts that you use with the Mail app on both devices? If so, does the same issue persist with those email accounts as well?
We'd also recommend testing to see if emails as syncing as expected if you sign into your webmail account as well. Lastly removing and readding the account can refresh the accounts connection, which may be able to help with this issue.
If you can’t send or receive email on your Mac - Apple Support
Test using webmail
Many email providers offer webmail, which is a way to use email with a web browser such as Safari, instead of with an email app such as Mail. For example, if you're using an iCloud Mail account, you can use iCloud.com to send and receive email.
Webmail is a good way to verify that your account is valid, but it doesn't mean that your account is set up correctly in Mail. Your email provider can help you verify your email settings and check for other issues that affect sending or receiving email.
Remove your email account, then add it back
If no other solutions work, follow these steps:
1. Remove your email account in Mail.
2. Add your email account in Mail again.
3. Try to send and receive email.
Keep us updated on the results.
Have a great day!