They're similar but a little different. I'm from the U.S. so I probably think differently than someone from England or Australia, but these are my feelings.
- We use it more often than "rely on me", so the meaning usually isn't as strong as "rely on me"
- "You're going to like this restaurant. Trust me."
- "I'll definitely finish my chores tonight. Trust me."
- The meaning is similar to "believe me. This is true"
- "I'm not lying! Trust me"
"Rely on me"
- Usually stronger than "trust me"
- "You can rely on me boss. I'll finish this work by tonight."
- There is usually a task. Something must be done
- "My mom always relies on me to help her with the laundry."
- Also, it feels like the person is necessary. "I'm relying on you" = "I need you"
- Can also be bad
- "He relies on others too much. He should work harder."
- "My boss always relies on me to clean the office. I'm busy too!"
Here's an easy rule:
If there's a task--> "You can rely on me." (I'll definitely do it. Don't worry)
If you want to say "believe me"--> "Trust me"