Well I see why some would compare the two games as they don't relate in terms of placing on the time line and they're pretty similar in regards to design and perceptive - the only defining difference is the time frame.
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Anyway, it's hard to really define which is better for their time as there is such a big time gap in between the two, which affects visuals, gameplay, controls, detail etc. A Link Between Worlds to me though is the pinnacle of top down Zelda; it takes specific elements of what makes the 2D games great, combines them, and--in a way--perfects them. ALBW Is almost a flawless game, I honestly can't think of any stand out bad design choices. Sure it was a little on the simple side, but it was simple in an extremely fun way -- this game is just the that: incredibly fun.
Link's Awakening Is decent in its own right. The unique setting of a tropical island, challenging and mind boggling puzzles, a nice break away from the cliché story, and the underlining hidden story that's quite sad in a way. Of course age still plays a part, it doesn't hold a candle to ALBW's unique style, lovely visuals, and an incredible depth created by the 3D effect. Age also plays a part in it's challenge too. Navigating is a task while moving around is seamless in A Link Between Worlds. So when you do compare older titles to modern titles--challenge wise--age is vital to take into consideration. Now developers focus on making the game as flawless as possible and making the game accessible. Such time and effort into these things couldn't be achieved back then, so I would definitely definitely argue that the only reason LA Is more challenging is ultimately because of the time gap in between the two games.
If we compare the two games further then I'm afraid A Link Between Worlds easily surpasses Link's Awakening. Both soundtracks are amazing, but the latter has a really cool, almost old British-like twist on A Link to the Past's famous and iconic music. Combat is not only more efficient (again time age plays a factor), but it'd also more tactical; you use multiple items in combat adding diversity, which ultimately gives a small sense of strategy. If we delve into story and narrative then both are quite brief even in terms of Zelda games. There is little plot during the the main game itself, but I think the main thing is, when ALBW chooses to introduce plot elements, it executes them to great effect.
I could go on further and talk about exploration, overworld design, dungeons, bosses, puzzles etcetera, but there is much easier way for me to determine which is the better game: enjoyment. We sometimes over analyse, over criticise, until we have more negatives than positives which in the end obscures our view of what the game is. I like the step back and say to myself "which did I enjoy more?". The answer is rather simple for me then, A Link Between Worlds. This game oozed enjoyment and it was just a pleasure to play. It wasn't maybe as epic as the main console titles, but the simple fact that throughout my 18 hours with the game I enjoyed every second, which is something I rarely say about a game.