When you are using compared to or unlike you are comparing two or more things.
First of all looking at your initial question, when saying something like
Unlike A, A is worse
Compared to A, A is better
you are starting the statement with comparing A but then you are comparing A with itself. For example
Unlike the summer, the summer is warm.
This would be interpreted to be that the summer is warm but then summer is cold.
When using compared to, you are generally comparing similar items.
Compared to the yellow bowl of soup, the red one is hotter.
In other words,
The yellow bowl of soup is hot, but the red one is hotter.
Unlike is generally used when comparing things which are very different.
Unlike the yellow bowl of soup, the red one is hot
would be interpreted that the yellow bowl of soup is cold, whereas the red one is hot.
To use the earlier summer comparison again, we could say
Unlike the winter, the summer is warm.
The reason why I have said these are generally the way these comparative phrases are constructed, is because there are exceptions. You can use either compared to or unlike when the compared items are usually the same or usually different. However you would swap the compared items and its comparison.
Compared to the shop's own brand, the named brand is usually more expensive.
Unlike the named brand, the shop's own brand is usually cheaper.