Today, rap is one of the top genres of music. Modern rap has it's roots in hip-hop of the past. At first glance the two may seem the same, but they're not. An example of old school hip-hop is a sub-genre known as boom-bap. The heavy bass drumline beating rhythmically in a simple loop is what gives this old-school hip-hop form that title. A good example is, "I Ain't No Joke," by Eric B. and Rakim.
On the other hand, new school rap is more syncopated. It's sort of the way jazz is to classical music: though there may be the same notes, classical music follows strict rhythmic rules and jazz has no such boundaries which makes it free for syncopation even though it keeps the beat.
New school rap, having developed from old school, evolved and became more liberated with rhythm while still keeping the beat; today's rappers are less afraid to stray from ending every bar right on rhythm and play around more with the syllable count on each bar.
Another difference is the beat itself; as opposed to boom-bap beats, most modern hip-hop beats are heavy in bass and in up-tempo liveliness. A key aspect of trap music (a popular sub-genre of modern hip-hop) is the rapid burst of light hi-hat taps. These two elements combine to make new school hip-hop more unpredictable. A good song to exemplify this is "Vice City" by Black Hippy.
There are great artists in both modern and past hip-hop genres. Here's a list comparing today's artists to their predecessors.
Future and Dr. Dre.
Neither are good lyricists, yet both are great emcees, their tracks are hype, they both use great features, and make for better features themselves.
Metro Boomin and Lord Finesse.
These two great producers only produce for a handful of artists. They both have a consistent feel to their beats even though their styles are very different.
Kendrick Lamar and Nas.
Two of my personal favorite artists of all time. They're both conscious artists, but they also convey their gritty histories and violent pasts. Each of them are relatable to their audiences as they strive to be. They have good (not great) flows, but they work beats very well, are wordsmiths and, thus, are unafraid of proving themselves to the rap game.
YG and Snoop Dogg.
I'm not saying YG is anywhere near as good as the legendary Snoop Dogg. Snoop Dogg is immortally iconic to rap, whereas I wouldn't even call YG a veteran rapper yet. However, I do believe he has potential to do great things. He's similar to Snoop Dogg in that he has a very West Coast feel, with his flow, vocabulary, and beats. He's a great emcee, and can make songs ranging from scary to grave to funny.
Dave East and Biggie Smalls.
This is another another comparison that's difficult to make, just because The Notorious B.I.G. is in a class of his own. What's worse is that Dave East has only several years under his belt as a rapper, so comparing a Great to a freshman is mostly speculation. Dave East, like Biggie, has an entirely East Coast feel, and can surely be called street rap. He also has gritty lyrics, but unlike some modern rappers' lyrics, his are not glorifying the lifestyle which Biggie coined.
Desiigner and KRS-1.
These rappers have a lot in common when you get down to it. Although Brooklyn artist Desiigner is new on the scene, his two songs are upbeat, street-based, crowd-oriented, and perhaps most notoriously, hard to understand - all attributes that KRS-1 also has. Neither are afraid to do a bit of tasteful singing as well.
Mac Miller and Eminem.
No artist that I've heard can compare to Eminem's talent in unorthodox beat mastery, cleverness, emotion, and of course, wordsmithing. It's worth mentioning that multiple personas is a rarity throughout the whole music industry (especially rap since rap is so ego/attitude based - you want to be yourself). Eminem does this with his drug-induced Slim Shady. Mac Miller does this as well, with Larry Fisherman, Larry Lovestein, Delusional Thomas, and perhaps more. In my opinion, he does it better. Mac Miller also went through several vastly different phases in his life, as did Eminem, which noticeably seeped into their music. Mac Miller has potential to be just as great, exhibiting beat mastery, cleverness, and some decent wordsmithing of his own.
Delusional Thomas and Ol' Dirty Bastard.
A great of marginal rap is ODB, known for his drug-induced poetry. His and Delusional Thomas' drugged up rap is marginal because it's relatively slow, very chaotic, and almost never uses regular beats, yet the beats used are perfectly fitting. It's worth noting these two artists are underground, which is noticeable in their uninhibited music.
J. Cole and Big L.
These two are artists that can make you nod your head 'til the end of time. Each have fantastic punchlines that will make you laugh in the middle of a track, and each are great emcees. They also have great beats that any artist would be happy to debut.
The Game and Big Pun.
Big Pun and The Game are an unlikely combination, but for their flows, darkly gritty lyrical content and respective street cred these two are comparable.
Chris Rivers and Papoose.
Papoose, the underground legend, is the embodiment of the East Coast feel. As much as I wanted to place Chris Rivers by his father, Papoose is a better match for the Eastern flow. They also share a frequency of punch lines and a knack for having catchy hooks.
G Herbo and Mobb Deep.
G Herbo may not seem like an easy comparison because of his youth. On the contrary, Mobb Deep also debuted in their teens. From the few songs out by G Herbo, I gather that he is a true street rapper, and had similar timing on his bars to Mobb Deep, relative to their beats. G Herbo could do well with their incredible ability to deliver.
Action Bronson and Ghostface Killah.
All jokes aside, if I didn't know any better I'd think these two are the same rapper.
Meek Mill and DMX.
If attitude, pride, and confidence are the qualities of a rapper that have you hooked, these two are the rappers to go to. On another note, I don't think either can rap without yelling. What?
Chance the Rapper and Tribe Called Quest.
Chance the Rapper is so unlike most other rappers because he refuses to make songs that aren't positive in some way. Another oddity is his use of instruments in his songs, Donnie Trumpet is featured in a great many of Chance's songs. Before the violent, gritty movement of '90s rap, all rap was nonviolent. Tribe Called Quest is a group of that era that Chance probably belongs to at heart.
Nicki Minaj and Lil' Kim.
Nicki Minaj sold out a long time ago. Her underground hip-hop career is not to be discarded though, she had the potential to be a different kind of famous, following in Lil' Kim's style as well as her footsteps, until she sold out.
Joey Bada$$ and Cassidy.
Joey Bada$$ is only growing his popularity and skill. I think he'll be the next Cassidy, with great rhythm, flow, and funny punch lines. Cassidy is known for his diss tracks, and although Joey Bad hasn't made too many enemies lyrically, his diss tracks are great. What can be called his 7 Fingers of Death was recently released on Sway's Universe, aimed at Troy Ave.
Montana of 300 and Lil' Wayne.
Lil' Wayne may still make music, but some of his albums can be considered non-current. Those albums contain the aspects that are comparable to Montana of 300. They include frequent and silly punch lines, street feel, somewhat street songs, and slightly short bars.
Danny Brown and Busta Rhymes.
Busta Rhymes is one of the quickest spitters in the game. Danny Brown is a very nontraditional rapper (as Busta Rhymes was in his day). They both have real quick paces, are hard to understand, are tremendously emcee oriented, and can have at least one silly bar per verse.
Immortal Technique and 2Pac.
Although not overly relevant, I think Immortal can be considered fairly modern. 2Pac, like Biggie, Snoop and Nas on this list, is forever a Great of rap. What makes Immortal Technique like 2Pac is that only those two make violent and gritty music as well as visionary music.
I'm leaving out tons of rappers and producers, since I can't speak to music I don't listen to. Feel free to comment who those artists are, and which artists of the past and today compare to them.