Like everybody else said, I would go for New Leaf. I got the game two or so years ago and I still play it regularly. It's my favourite 3DS game.
In New Leaf you are the mayor of your own town. As the mayor you are given the ability to "design" your town. With restrictions: Villagers slowly recommend you projects you can build, you can only have so many projects in town, certain projects can only be built once, or like bridges there is a maximum of three I believe, and Isabelle had to OK your placement. You can plant various trees, bushes, flowers (there is also flower breeding to get special colours) and bamboo shoots. With the help of the QR code reader you unlock in the game you can place "paths" along the ground. You can either make these paths yourself or get them online, via QR codes. In Happy Home Designer you haven't got a town, you have houses. I believe there is a set number of houses or buildings you can design a day, but I'm not sure.
When you first start the game you answer a series of questions which determine what your face looks like. You can google specific answers for specific faces. When Shampoodle opens you are able to change your hair style and colour, as well as your eye colour. Alternatively you can just use your Mii head (note that playing with your Mii head disables the use of head accessories I'm pretty sure). I think in the summer time you can get a tan so your skin is darker for a short amount of time, however it's not permanent and the game gives you white skin - this is not a choice. I believe Happy Home Designer gives you much more freedom when it comes to choosing what your character looks like.
In terms of clothes, the game has quite a bit to offer. The game has stores that sell clothes (shirts, skirts, pants, dresses), accessories (hair pins, umbrellas, hats, masks, etc) and shoes. You can also design your own shirt (long sleeved, short sleeved, or sleeveless), dress (long sleeved, short sleeved, or sleeveless), a hat (two options, I forget what they are though, my apologies!), and an umbrella. Or you can get other people's designs via the QR code reader. I believe in Happy Home Designer you only have the one outfit, but I'm not sure.
You also have a house. You can pick how many rooms you want and there are three different sizes the rooms can be. You can decide how the outside of the house looks. The game offers different options - it is not like building a house in The Sims. You are given a default layout every time. The rooms are always square. Just a bigger square when you upgrade them. You can have a max of six rooms I think? You can pick wall paper and flooring for every room and you buy/collect furniture for them. Some items you can interact with, but it's generally just like, "turn on/off the light, turn on/off the radio/tv, hear the typewriter make noise and move" just little interactions. You can't cook or actually run a bath or anything like that. Happy Home Designer likely offers more in home design, as that's what the game is about, and there are more interactions.
New Leaf also has holidays which are celebrated. I think all the holidays have a little game or something to go along with them. These are celebrated in real time, they tend to follow the U.S's holiday calendar (so Thanksgiving is in November) and the holidays are Toy Day (celebrated on Christmas), New Years (celebrated on New Years), Halloween (I think it's given a different name, but it's celebrated on Halloween), Easter, and I feel like I'm missing one. Plus there are different contests that appear.
You can fish and go swimming in the game, as well as collect bugs and fossils. There is an encyclopedia that keeps track of what you've caught in terms of fish and bugs (fish you catch through swimming have their own encyclopedia) and the fossils as well as paintings are donated to the museum.
There are some mini games you can eventually play - I find they are more enjoyable when played with friends.
There is also multiplayer. You can visit other people's towns. This allows people to trade items, invite villagers to move in, or play mini games together. There are also "dream worlds" where you just visit the town. The other person isn't actually playing with you and anything you do while visiting their dream world isn't permanent.
I may be missing some things but that's basically New Leaf. It's a nice open ended game. You can shape your town/house so it suits you.
You also have villagers, of course, a maximum of 10 villagers and where they move in is random - that means that they can crush flower beds. There is a work around but it's a bit tedious.
I've not played Happy Home Designer personally, so the bits I said about the game are just things I've heard or seen in gameplay at one point or another. What I said may not be completely accurate.