It's heuristics, not grammar. You usually appreciate a noun or noun phrase in a transitive sentence (X appreciates Y).
Case 1: If it's something useful (e.g. 'your help', 'what you did') then it's more likely about being thankful.
This use of 'appreciate' is also a social action. You say it to make someone feel good for helping.
Case 2: If it's an abstract thing (e.g. 'the importance of', 'the difficulty of') then it's more likely you're talking about understanding.
Usually when you talk about this usage it's because the _fact_ of understanding is important to the speech act. If someone explains the law of gravity you don't say "I appreciate that", you say "I understand that". But if you fell from a height and hurt yourself you can say "I didn't appreciate the law of gravity".
Case 3: If it's intransitive and it's about money ('the car appreciated in value', 'the house appreciated in value').
It's a pity that heuristics are not mentioned in most language books or dictionaries. At least I wasn't taught any in school (10+ years ago). This should be next to the word definition.
Case 1: If it's something useful (e.g. 'your help', 'what you did') then it's more likely about being thankful.
This use of 'appreciate' is also a social action. You say it to make someone feel good for helping.
Case 2: If it's an abstract thing (e.g. 'the importance of', 'the difficulty of') then it's more likely you're talking about understanding.
Usually when you talk about this usage it's because the _fact_ of understanding is important to the speech act. If someone explains the law of gravity you don't say "I appreciate that", you say "I understand that". But if you fell from a height and hurt yourself you can say "I didn't appreciate the law of gravity".
Case 3: If it's intransitive and it's about money ('the car appreciated in value', 'the house appreciated in value').