They do mean the same thing but search has more of an effort feeling to it. Like if you were to say "I am going to look for my hat." It gives the idea that you have not spent much time looking. If you were to say "I am going to search for my hat" it gives the idea that it is lost and you are going to put a lot of effort into finding it. Another example is a Google "search": Google has to look at a lot of websites to find the best match to your keywords.

In most cases they mean the same, I'm searching for / looking for a new plant vase.

Look for is a phrasal verb that basically means the same thing as search.

They both mean the same thing.