A port is basically the same game created for one system but fit onto mulitple systems, while "remakes" involve the developers re-creating the game from the ground up. Remakes are made many years after the original game. One of the main reasons developers remake games is because the advancement of technology allows them to fully realize the experience. For example, an old NES classic game may perform better when remade for newer hardware giving it a cleaner look and tighter gameplay. Remakes can lead to an amazing new experience that enhances the original game. Here are some examples how remakes that were done extremely well and even replaced the original game as the definitive version.
Super Mario All Stars
One of the earlier remakes in console history was done by Nintendo. The SNES was out and a huge success. It was such an improvement over the NES (control, gameplay, graphics, and sound) that it seemed like a smart move to upgrade the original Super Mario Bros games. They packed Super Mario 1, 2, and 3 along with the "Lost Levels" (which was Super Mario 2 in Japan) all onto on SNES cart. The original games already had tight controls so only minimal tweeking was needed in regards to gameplay (they re-did Luigi's controls to make him faster, a higher jumper, and also slightly slippery). The real improvement came with the new graphics, the levels were now more colourful and the characters had much more detail to them. Another huge improvement was the animation, the NES just could not handle certain animations that the SNES could. The new animations gave the game a much more fluid look to it and the characters now moved more realistically. Super Mario All-Stars is now the best way to play the clasic Super Mario games (even the GBA ports used All-Stars with minor tweaks to fit the handheld). For a time Nintendo even bundled All-Stars plus Super Mario World with the SNES.
The first Resident Evil was a ground breaking title for the Playstation. It brought in the survival horror genre that still thrives to this day. However, the games graphics did not age well with time. As players became used to more realistic graphics the blocky characters in Resident Evil no longer frightened players. Resident Evil also leaned heavily on puzzle solving. These puzzles were fun to figure out, but like most puzzle solving games once the player understands how to solve them the second play through becomes much less rewarding. Finally, the original Resident Evil game suffered from a poorly translated script and cheesy dialogue.
The Gamecube remake of Resident Evil solved the problems the original game faced. The graphics were upgraded to a level that looked as realistic as the Gamecube could pull off (and still looks fairly good today). The mansion you explore was now redone with new rooms and puzzles that challanged even the most advanced Resident Evil players. The script was also rewritten and the revisions made it a lot less chessy and the grammar was more sound (yet it still maintained its B-movie atmosphere). In some ways Resident Evil was ahead of its time and it wasn't until the gamecube remake that people got to see what the game was capable of.
The original NES Punch-Out (titled "Mike Tyson's Punch-Out" before Tyson got into legal troubles) was one of the first games I ever played. The NES game featured larger than life boxers that you had to fight your way through. Each fight was like an epic boss battle, and they got progressively harder. The NES game had near perfect controls that had the player thinking in fractions of a second in order to dodge their oppenents. It was a fast paced game with a lot of character and charm to it. The NES cart had to have a special chip inside of the cart to help render to giant characters.
As I am writting this the main game I am playing is Punch-Out for the Wii and I am completely hooked on it. It has everything that made the original game so great while at the same time improving it in all the right areas. The NES game is a masterpiece and its core gameplay that got so many gamers interested remains. The Wii game took the core gameplay of the NES game and only built on it slightly (adding a duck to the dodges as well has a more robust counter punch system). The giant boxers you faced in the original game are now bigger, more colourful, and have a larger moveset. The boxers also have more personality than before. The NES game had to rely on character design and text between matches to get across their traits (which actually worked very well). This is fleshed out even more as the boxers now have full voice and much more fluid animation. The smartest move that this remake made was to cel-shade all the characters. The boxers personalities and character designs lend themselves to a cartoon world. If it was remade in realistic graphics the game would have looked ugly, and lost a lot of its appeal. Also have the game remain cartoony allows it to be timeless (cel-shaded 3D cartoon models age much slower than realistic graphics). It was a real feat to capture everything that made the NES game so great over 20 years later and still improve on it.
Final Special Mention
The last thing I am going to talk about is somewhat of a hybrid between a port and a remake. During the early console days it was near impossible to port over an arcade game; therefore a new game would have to be made for the console. This usually led to a very different experience on the console. The most notable home of unique arcade port/remakes is the NES.
I love that I can mention Punch-Out twice in the same post. The original aracade game only featured a few boxers (only Glass Joe made it to the NES game). It is one of the best examples of the NES game being much deeper and fully realizied even though it was on inferior hardware to the arcade. The arcade Punch-Out was a fun game but it just didn't have the same level of charm that the NES game did.
Ninja Gaiden was originally an arcade brawler. It really did not stand out against the other hundreds of brawlers in the arcades (even though it was a solid game). The NES game ditched the slow brawl for a fast paced action platformer. The NES game was one of the hardest games ever created (while still being playable and fun). It required quick reflexes and mastering the surroundings. Ninja Gaiden on the NES felt a lot more like playing as a ninja than in the arcade