My opinions of both these games have fluctuated wildly over the years.
Skyward Sword impressed my the first time I played it, but I think the game has terrible replay value, and has fallen for me upon every subsequent playthrough. Twilight Princess, on the other hand, felt like an Ocarina of Time remake when I first played it, but over time I have grown to really appreciate the game and what it brought to the series.
First, I'll talk about characters, because this is something I think both games struggle with. Twilight Princess has some great characters; Midna, Zant, Malo, Zelda, The Postman, Telma, to name a few. But it also has its fair share of really bland characters, and the ones that should be interesting (the resistance crew instantly comes to mind) just fall short due to lacking much personality and not really standing out amongst each other. Then there are the gay stereotypes that run the cannon minigame in Lake Hylia who are just creepy.
In the case of Skyward Sword, the standout characters are Ghirahim, Gorko, Batreaux, Groose, maybe (and reluctantly) Fi, and that's really about it. It suffers from the same problem in that characters that should be interesting, such as Impa or Gaepora, just end up being rather bland to me. Additionally, the character design in Skyward Sword is just downright awful. The vast majority of Skyloftians look horrendous, and there is something about Link's face that really makes me want to punch it. So in terms of characters, I gotta give the edge to Twilight Princess.
Next, gameplay. This one is much easier for me.
Twilight Princess has top-notch gameplay. Not only is the swordplay - with the intense sword-sword combat in the game and the many Hidden Skills making for a highly diverse move set - the best in the series bar none, but also the items, limited in use as some of them may have been, are among the most inventive we've seen yet. From bomb arrows to the ball & chain, and then the double clawshots, the game went all out on its items. Yes, many of them do suffer from a certain amount of uselessness past the dungeon they are received in, but that does not make them any less innovative and interesting. Additionally, Twilight Princess is home to the best horseback combat in the entire series. Then we get into Link's Wolf form, which shy of unique attacking methods as it is, Wolf Link nevertheless is a really cool and unique aspect of the gameplay that makes Twilight Princess stand out. My main gripe is that the motion controls in the Wii version are clunky and feel tacked-on (which they were); the gamecube version is far superior, and the gamepad for the HD remake also feels nice.
Skyward Sword was developed with motion controls in mind. As such, the entire game, from the gameplay to even the story, revolves around the motion controls. I never really had a problem with them myself, but I do wish they were optional. In any case, how does this game compare to Twilight Princess? Looking at the swordplay, while the motion controls do make the swordplay unique and fun, there really isn't much to the combat beyond swinging in the proper direction at the right time. In fact, that's all it is. It was a neat idea, and certainly fun, but after the first couple hours of the game it just got boring. The items are also not that interesting in Skyward Sword, with the exception of the Beetle, which is also the only truly unique item in the game. Did the game do a better job utilizing its item pool than Twilight Princess? Sure, for the most part I can concede that. But its relative lack of innovation and originality puts it a step behind because the items just aren't that interesting. I do appreciate being able to roll bombs, however.
So, from a gameplay perspective, that is another point for Twilight Princess.
Next, I'll discuss the stories. This section will probably be comparatively short.
Both Skyward Sword and Twilight Princess have fairly simple and typical stories; people close to Link get kidnapped and Link must rescue them, discovering himself as the chosen hero on the way. It is a concept that is used to death in Zelda games at this point. I have always felt like Twilight Princess had a slightly more immersive story, though I know this can be a rather polarizing opinion. I like the nitty gritty of both stories, as both are very personal for Link and feature some dark themes and emotional moments. However, I have to give the edge to Twilight Princess for telling not only Link's story, but I would argue the story as a whole is more Midna's story than Link's, which is a nice change. Additionally, I feel more attachment to the Ordon kids than I ever did to Zelda in Skyward Sword as I never found her that interesting and, as a result, never felt much of a sense of urgency. As a much larger minus for Skyward Sword, for a game that was supposed to be a prequel to the entire series, it takes a few good measures to shatter the established lore of the franchise.
Dungeons are one thing I will endlessly praise both games for. I think both Twilight Princess and Skyward Sword did an excellent job in giving us interesting, well-designed dungeons that are more than just an elemental Temple. From a mine in a volcano to a mansion in the mountains, a buddhist-themed cistern, a pirate ship, etc., both games excel at creating some of the most creative dungeons in the series. It is difficult to give either one an edge as there are very few dungeons in either game that I dislike, so I am going to give it to Twilight Princess simply because there are more of them. Skyward Sword definitely has a fantastic batch of dungeons, though.
The last topic I will discuss here is art and music.
In terms of artstyle, I will be blunt; I don't think either game looks astonishing. Twilight Princess is too dark and lacks colour, just being presented as rather bland, while Skyward Sword is a bit too bright. The latter is more colourful, and I feel like its artstyle does stand out a bit more and gives the game more of an identity than Twilight Princess's does. I especially love the way Eldin Volcano and the lava areas look in Skyward Sword. So, for art, I will give the edge to Skyward Sword even though I don't think either look particularly great.
For music, it is a tough call. Twilight Princess has far more memorable tracks, but it is hard to beat Skyward Sword's fully orchestrated soundtrack. It makes the game come to life more in my opinion and it has some really fitting area themes, even if they lack in memorability. Additionally, Skyward Sword has one of the largest music libraries of any Zelda game, even possibly coming in at number 1 due the fact that many locations have more than one theme. I find myself listening to the music of Twilight Princess more, and in terms of which soundtrack is best outside the context of the game, I would have to give it to that game immediately. However, with the music of Skyward Sword being orchestrated and so fitting for most areas, I will give it a slight edge. So, in the art and music department, Skyward Sword wins.
Other areas I considered discussing are just too close. For instance, both games have slow openings that often feel like a drag, but incredible finales; both games have a fantastic villain in the form of Zant and Ghirahim and a significantly weaker "main" villain; both games feature a good overworld with nice environments (though Skyward Sword would definitely lose some points for only having three themed areas and each of them being disconnected from each other); and both games are genuinely fun experiences when it comes down to it. In my own opinion though, Twilight Princess is just a better game, both within the Zelda series and the video game market as a whole.