Translation:The neighbor's windows are dirtier than ours.
Since we're talking about some neighbor's windows, we need the article because "neighbor" is a noun. In the other sentence that you mention, "the" is not needed, because "František" is not a noun but someone's name. "The František's dogs" would make no sense in English.
There is (as I write...) no Reply button for your last comment, so I'm replying here.
In the new examples that you give -- "mother's beer" and "father's car" -- "mother" and "father" are being used as proper names rather than simply as nouns, even though they are not capitalized.
In such cases, the possessive pronouns are not needed. Those sentences are treated just like the one about "František's dogs," where the article is not needed. I hope this helps, at least a little.
Just to add to the confusion... At least in the US many people actually do say, "The wife's...etc." But in this case, "the" really means "my"; it's quite a colloquial usage. :-)
Apart from that specific case, "the" would be incorrect. Normally a possessive pronoun is used: commonly "my," "your," or "his."
@BoneheadBass, @VladaFu .. lingot for both of you for trying to help. But still ...
There is a fly in mother's beer => could be without possessive; I like everything from MY mother's kitchen => MUST be MY
What would you do without father's car => could be without possessive; I am afraid of MY father's cousin => MUST be MY
Why is it like this?
British spellings are normally treated by Duolingo centrally. If they miss it for neighbour, it is good to complain to them in the main forum.
We can add it to our sentences manually, but it is not what we are normally supposed to do.
It seems there is something more going on and we will try to investigate.