I have a feeling that one might mean slightly earlier. Is this right?
As pointed out by xiaohouzi79, 晨 actually carries the meaning of sunrise, hence implying early morning. However this indication is weak, especially when you contrast 早晨 and 早上。
However 早上 would cover the entire morning, say, from 6am to 11:59am. Therefore in a way, it is less specific.
If we want to refer to extremely early morning, before dawn, say 3am to 6am. We sometimes use the word 清晨 and if we are talking about "morning" as in "after midnight" (00:01 to 3am for example) we usually use the word 凌晨 .
So after sunrise, we usually just say 早上 , since that time period is not close to 晨 (sunrise) anymore.
Generally speaking, people will treat 早晨 and 早上 exactly the same way unless explicitly directed other ways. It's OK to translate both of them to morning.
In case you want to express early morning or daybreak, you may consider using 清晨.
I agree with Terry that 早上 and 早晨 are synonymous, with perhaps very slight differences in connotation. Pace Gapton, I wouldn't say that 早上 can always refer to any time from early morning to 11:59 am; for that time period, in some cases it is better to use the more general term 上午.
One place this distinction is important is in greetings. A person writing an email to a colleague at 10:30 am wouldn't start their email with 早上好 ("good morning") because it sounds like the writer just got up. In this case, the looser term 上午好 is more appropriate.