"I don't really think he likes our food."
Translation:אני לא ממש חושב שהוא אוהב את האוכל שלנו.
So באמת and ממש are different but the English sentence could be translated either way.
באמת = truly/really (lit. in truth)
ממש = really
So your sentence would mean "I don't actually think that he likes our food." Which is one of the possible meanings you can get from the English sentence: "I don't really think that he likes our food." The other meaning in English would be that you have some doubt to whether or not he likes our food. This is the meaning that is conveyed with ממש
Probably for the same reason אני ממש לא חושב שהוא אוהב את האוכל שלנו was not accepted? I always thought that the לא had to go in front of the verb, but apparently it goes before the adverb and the adverb goes before the verb. Also, באמת means really (in a literal sense, in truth or agreement) and ממש means really (not so literal, more a sense of "very").
Thanks for you answer. I still think Duo's not accepting this is strange. The two words may have slightly different meanings in Hebrew but in English they are too close to call, so both should be acceptable in an exercise like this, in which we are using words out of context. In fact, Duo itself uses ממש in similar example in another exercise, so there's an inconsistency here. Also, I'm not sure that the word order here in Hebrew is so rigid that a very slightly different version would actually be 'incorrect'. I still think Duo needs to be a bit more flexible in this case.