- There are tools for cracking file encryption passwords, including PDF. Because it's using AES-256, which is designed to be fast, it's possible for someone to do dictionary and hybrid-dictionary attacks on the file at paces of millions per second. By using a public service, you allow anyone with access to mailboxes, such as Google employees, to try their hand at an offline attack. If the PDF is cracked, you will never know.
- You lose deniability that you sent the message. The subject, body and headers of the MIME encoded mail all reveal details of the mail that could leak information about the contents and intention of the message.
If you must send an encrypted PDF to a recipient, don't use a protocol where a centralized server can act as a middle man. Instead, use a peer-to-peer client, where the only possible way to intercept the file is sniffing packets on the routers and switches in the path. Further, make sure the peer-to-peer client software supports client-to-client encryption. Even though the PDF may be encrypted, you will want to encrypt the rest of the IP packet data, to prevent leaking formation about the contents of the packets.
However, the ultimate way to transfer files securely is using an air gap. In other words, the Internet is never involved. Then faceless government agencies cannot even attempt deep packet inspection and cracking at the encrypted packets, that could even be generated with software that the agency has a backdoor to. Instead, create the file, encrypt it, transfer it to a USB stick, and physically give that copy to the recipient in question.