The latest security flaw was reportedly found by Telefonica consultant Jaime Sanchez, and it would allow a malicious user to essentially reuse old message tokens to send new messages. Using this method, someone could potentially flood your inbox with hundreds or thousands of messages. As yet, there is no evidence that the flaw has been exploited to cause actual problems for anyone. But, when Sanchez demonstrated the attack for a reporter at The LA Times (video at the source), he was able to send 1000 messages to the account in about five seconds. This kind of attack will cause an iPhone to crash and reboot. Androids could fare better depending on the specs, but it will still cause a major slowdown.
Even worse for Snapchat, Sanchez said that he hasn't even bothered reporting the flaw to Snapchat itself because of how the company responded to earlier reports of security flaws. Essentially, when earlier flaws were reported to Snapchat, according to Sanchez, "Snapchat didn't care."