First of all, if the traveling twin came to another planet and stays there he would not age faster or slower any more. Both twins would age at a same rate of 1 year per year. However the traveling twin would remain to be a little younger, age difference would remain constant from now on.
It's possible to make sure there is an age difference even without going back to Earth. (F.e. the one of the twins could send the other a message like "now I am 20 years old", as soon as message received the other attaches his age to the message "I am 20 years 20 days old" and sends it back, and so on. The message would grow in size and eventually would look like:
1: I am 20 years old
2: I am 20 years 20 days old
1: I am 20 years 30 days old
2: I am 20 years 50 days old
and from this list we can see that "speed of time" is identical for both twins but one of the them is older than the other.
The "return back to Earth" step is used to make the paradox more obvious: here are both of them, how come this one is younger, why not the other one?
But paradox remains even if they do not meet ever again.
Actually paradox starts here: we have two twins traveling in opposite directions. And from the point of view of each them the other twin is aging slower! Isn't it weird? Theory of relativity says that's ok. Sounds like nonsense. And it's a very natural attempt to construct a paradox to make this nonsense obvious. After all it's impossible that each of the twins is aging slower the other, isn't it? Let's bring them together and just look at them. Each of them expects the other to be younger, but that simply can not be, so the theory of relativity is, well, not correct!
But there was a mistake here. The mistake is done by traveling twin. Yes, from his point of view his brother was aging slower than him. Almost. Something very interesting happened during the "turning back". Let's assume the acceleration was very high, so the whole turning back took a very short time.
Nothing special happened to the traveling twin during this acceleration moment. Neither from his point of view nor from the point of view of his brother. He did not become younger or older during the process. Something strange happened to his home-sitting brother during the acceleration!
Suppose it took several milliseconds to fully stop his spaceship. During these milliseconds the earth suddenly became much-much farther from him. (the imaginary ruler stretched from the Earth in his direction is staying still now and thus the is no length-shortening effect). And time on Earth advanced for several years! Now he accelerates to full speed, the Earth becomes much closer again and time on it advances several more years.
From now on time on Earth goes slower than his time, but as a result of these "jumps" during acceleration when he comes home he would see that his brother is older than him.
Once again the most important point: nothing special happened to traveling twin during acceleration. Something special happened to the rest of the world from his point of view.
So, the relativity theory says: no paradox here. Traveling twin will be younger. When you tried to describe the situation from his point of view and show that the home-sitting brother will be younger you made a mistake.