While the two phrases seem quite similar, there is a big difference between the two:

Transnational Crime can be defined as “criminal activities extending into, and violating the laws of, several countries”, for instance: Money laundering, Illicit drug trafficking, Computer crime, Terrorist activities, Corruption/bribery of public officials

International Crime can be defined as: “crimes violative of international laws including but not limited to crimes against peace and security of humankind”, for example: Genocide, Crimes against Humanity, Torture or Slavery.