When comparing the two in a same language, there is no difference in lyrical choice from "rap" or any other musical form in word choice or topic. The phenomena of dropping letters from end of words is not unique to rap and can be found in non-rap lyrics as well, such as the words of the U.S. National Anthem:
O say can you see, by the dawn's early light,
What so proudly we hailed at the twilight's last gleaming,
Whose broad stripes and bright stars through the perilous fight,
O'er the ramparts we watched, were so gallantly streaming?
And the rocket's red glare, the bombs bursting in air,
Gave proof through the night that our flag was still there;
O say does that star-spangled banner yet wave
O'er the land of the free and the home of the brave?
PLAY BALL! Sorry, Sorry, Force of habit.
Dropping letters in lyrical compositions is (and non-lyrical works) is called "Contraction" and is designed to evoke a more spoken language sound (English and any language really, is rarely spoken grammatically correct. One of the reasons Pompeii is so important is that the volcanic explosion preserved the city's graffiti giving scholars great insight into how ordinary Romans spoke Latin (called Vulgar Latin to distinguish from proper Latin). English is prone to dropping letters not critical to the word's sound when it needs to move fast. Most speakers will use a "Contraction" in dialog.
In lyricism and poetry, contractions are frequent because, in addition to being closer to human speech, lyrics have to fit a certain pattern of beats, stresses, and other complicated patterns associated.. The above demonstrated "O'er" is quicker to say than "over" and The Star Spangled Banner is already notoriously difficult to sing.
Rap derives from music and poetic styles that originated in the African-American community and will often reflect slang used in those dialects. However, they are not always indicative of that community's slang. Your example of dropping the letter "g" from "-ing" words is something that is common in my very rural mid-atlantic community as well as many southern communities in the U.S. It wouldn't be a stretch to find a country song that does this too. Dolly Parton's "9 to 5" comes to mind as an example (Workin' 9 to 5/ What a way to make a livin'/ Barely gettin' by/ it's all takin' and no givin'). More recently "Old Town Road" by Lil Nas X combined Rap and Country and used several dropped "-ing" words in it's lyrics sung by both a Rapper (Lil Nas X) and a Country Singer (Billy Ray Cyrus) to draw two seemingly distant genres closer (it also uses contractions like "Gonna" (going to) and "'til" (until) prominently in the chorus.