What's the difference in grammar, meaning and so forth between
There is no such thing like
божественная любовь, right?
There's difference, but it's minor, slight stylistic peculiarities.
Любовь Бога- Like every attribute expressed by the Genitive case it has 2 meanings, the active and the passive one, the former is "I love God", the latter is "God loves me."
Любовь Божья- This one means "The Love of God, the love of the kind as God loves"
Любовь Божия- This one is almost the same as #2, but it's more Church-Slavonic than Russian, it sounds the way they say it in churches.
Божественная любовь- this is rather different from the previous ones, it means "divine love" which is more about the quality of love than about the subject of love, it's not about who, but about how one loves.
Любовь Божия == Любовь Божья, but first one is old style, second one is new style. can be an answer for question "What makes you happy?"
Любовь Бога - can be answer for question "Whose love makes you happy?"
Божественная любовь - there is such thing, but usually this phrase does not have anything in common with God. Here "Божественная" is adjective that means something like "great", "awesome" etc.
For example: Эта еда была божественна! This dish was awesome!