Being such a classic game, SMB3 has been ported to future Nintendo systems various times.

Super Mario All-Stars (SNES)

The first port was in Super Mario All-Stars for the Super Nintendo. It was a compilation cart that also contained SMB1, 2, and the Japanese version of SMB2(The Lost Levels.)
Super Mario All-Stars was released in 1993.

In this port, the graphics were updated dramatically, as well as updated audio, and some other new features.
Super Mario All-Stars introduced a Save Feature into SMB3. Once you beat a world, you are able go back and select thay world to start on.
Super Mario All-Stars also allows you to save the items you have in your Item Box.

Nintendo also released "Super Mario All-Stars+Super Mario World" for the SNES. It was pretty much the same, except it also included Super Mario World.

Super Mario Advance 4 (GBA)

The next port was 10 years later, in 2003. Super Mario Advance 4 for the Game Boy Advance.

This was a very well done port of SMB3. Nintendo didn't just take the SNES version and throw it on a GBA Cart. Once you beat the game, you're not only allowed to select which world to go to, but are able to select any level.

You are able to save a couple replays from the game, which is a neat feature. Saving replays was a fairly new thing back in the day.
Unfortunetally they made SMB3 a little easier. They made some platforms bigger and some things move slower.

A unique feature of this game, is its connectivity to the E-Reader.
There are 3 types of E-Reader Cards you could get.
First there are power-up cards, to get some items.
Second there are demo cards, which allow you to watch demonstrations of levels from
professionals. I guess that's pretty cool.
Now the third, this is what makes the e-reader feature amazing . . .NEW LEVELS!!!!
There are brand new SMB3 levels to play. So if you've played the original over and over and over, you can finally play new levels.
Unfortunately the E-Reader attachment was a fairly big flop, and now the e-reader cards for SMA4 are fairly rare, and aren't very cheap on ebay.

Virtual Console (Wii)

The Wii was released worldwide in 2006, and a big part of the Wii was its virtual console, which allowed a person to buy and replay classic games. So of course it wasn't long for SMB3 to show up in the Virtual Console.
Almost a year after the Wii's launch, SMB3 was made available in the Shop Channel.

This game plays just like the original on the NES. You can hold the Wiimote sideways and play it very much like a NES controller. The instruction manual is virtual on the Wii itself.
SMB3 was 500 Points, which is roughly $5, which is an amazing price for this classic.
The Wii allows you to stop playing NES games at any time, and come back and pick up right where you left off, a nifty feature.
I've played SMB3 many times on the Wii, although nothing beats the feel of playing it on the NES.

Virtual Console (3DS/Wii U)
SMB3 became available in the 3DS and Wii U's virtual console around the same time, late 2013/early 2014 (depending on the country).
It plays like the original for the NES. The Wii U and 3DS now allow you to create one save state.

The Future
With Nintendo beginning to enter the smart phone market, I believe it's only a matter of time now, that we'll see SMB3 available on an Android and iPhone.