The difference is that "to lie" is intransitive and "to lay" is transitive.
I lay the book on the table. I laid the book on the table yesterday. I have laid many books on many tables in the past.
I lie on the sofa. I lay on the sofa yesterday. I have lain on many sofas in the past.
The use of "down" makes no difference to the meaning and use of the two verbs.
ED. Yes, I forgot to mention: there are many threads on this subject. Thanks for the reminder, Dimcl.
Depending upon your degree of pedantry, the answer is:
1) The members of ABBA were not native English speakers.
2) Songs don't really need to obey the rules of grammar.
3) 84.683% of English speakers don't bother with the "lie/lay" distinction.
The term "lay down" is used improperly in the case of the ABBA song. But keep in mind that the term "lay down and die" has also become a frequently used term and could even be considered an idiom nowadays to mean "give up."
Remember in songs, the songwriter also must often bend phrases and words in a way perhaps not always proper in grammar but more "conversational" to convey a certain feel or mood.
In the example you give, the word "lay" is also an "inside rhyme" with the previous line's "May". This helps to break up the frequent rhyming sounds of try,might, die and I, so "lay" would be the better choice regarding writing the song.
So its more about taking poetic license with songwriting and not so much about being grammatically correct.