No native Arab speaker, but in this course at least, they use "-اَلْ" when it is the first word in a sentence OR when the previous word ends in a consonant, and "-الْ" when the previous word ends in a vowel.
When it is the first word in a sentence or when it is preceded by a consonant, the definite article is pronounced "al-"; when the word is preceded by a vowel, however, the "a" in "al-" is not pronounced, so the article is pronounced "l-". This course uses "اَ / ا" to make this distinction, but I am not sure how widespread this notation is.
For more info, search for "hamzatul wasl", which is the name of this linking "a" vowel.
They are the same, there is no difference at all. The first one have all Harakaat And the secind one, there is on Haraka missing. It is very common to write the words without Harakat You can write it like this المطر