I have never actually played any Master System games. What are the best ones to try?
Here are a bunch of SMS enthusiasts' choices from a recent thread at Sega 8-bit. http://shinobiman.proboards.com/thread/10628/best-games-celebrate-years-sms
If you're into "shmups" I recommend checking out R-Type, Power Strike, Power Strike II, Fantasy Zone, Fantasy Zone II, and Sagaia.
If you're into platformers, I recommend Kenseiden, Master of Darkness, Ghouls 'n Ghosts, the Sonic games (different from Gen/MD versions), the Mickey Mouse games, the Asterix games... there are tons really.
There are some good platformers with light RPG elements, like Wonder Boy in Monster Land, Wonder Boy III: The Dragon's Trap, Golvellius, Spellcaster.
There are also a clutch of fine RPGs like Phantasy Star, Ultima IV, Y's, Miracle Warriors, Golden Axe Warrior (often dismissed as a Zelda clone but a much deeper and larger game as one would expect since it came out 4 years later).
It's worth noting that, since the Master System didn't sell as well in the States, it was discontinued around 1990. But it continued going strong in Europe through the mid-90s. So a lot of the later European releases really push the capabilities of the system in amazing ways, as the programmers learned more and more how to squeeze every ounce of power out of it. At the same time they were making their code more portable so there were a lot of nice conversions from 16-bit Mega Drive games to 8-bit Master System games. Having said that, though, there is often a lot of surprising gameplay depth in the seemingly simple early games of '86-'89 as well...
As a kid in the US, I was lucky to have two other friends who also had a Master System, so between the three of us we got a bunch of different games that we could borrow and try out.
I would say one of the main philosophical differences between most NES and SMS arcade conversions was that the NES tended to have games that were re-interpretations of arcade games with new mechanics introduced to transform the games into longer experiences, while SMS games tended to try to reproduce the arcade games as closely as possible.
So for example, NES Double Dragon was changed into a single-player game that had a basic progression where you would have to unlock new moves as you progress through the game, while SMS Double Dragon was a two-player game with all the moves available from the beginning and the same progression through the levels. So ideally you could get better at the arcade games by 'practicing' at home on the Master System.
Of course there were exceptions to this general rule... for example, the SMS version of Ghouls 'n Ghosts was more the opposite, introducing a health bar and a shop for buying and equipping different weapons and armor and generally making the game slightly easier.
I think both game design approaches are valuable, and it underscores the fact that consoles used to be very different and distinct from one another, whereas today fanboys need to use magnifying glasses (a la Digital Foundry) to detect ultra-minor differences between the XB1 and PS4.