Thanks to all the members who have responded to my post. But doesn't appear mean seem?
Not in this context when you're referring to yourself. But, as I implied, it depends on the meaning.
1. "It seems [to me] like I slept for hours." (Referring to a feeling you have)
2. "It appears like I slept for hours."
Only 1 is correct.
3. "I seem to remember her."
Right "It seems that I remember her." Still right, but not good.
4. "I appear to remember her."
Wrong "It appears that I remember her." Still wrong.
If you are referring to a situation that you have evidence for (eg. someone told you that slept for hours, or that your surgery took only four hours), then you can say "It appears that I slept for hours", or "Apparently, I slept for hours" which are better in this context than "It seems that I slept for hours".