fit as a verb has sometimes the same meaning as an adjective. For example:
- the key fits the lock
- the key is fit for the lock
Can I use it both ways or there is some difference I should consider ?
Maulik V has more or less covered your original question, your new examples address the same difference in meaning but reverse the answer to "which is correct here"
The key fits the lock
is correct. Here fit means that the one item is physically the right size and shape to engage with the other item. It can also be used metaphorically, ie
He is a good fit for this job
The candidate's skills match up well to the job requirements.
X is fit for Y
is used to indicate suitability for a task or application. For example,
The water is fit for drinking
says this water is free from germs and poisons and may be drunk safely. As a UK native speaker I would be more likely to say
The water is drinkable or This tap provides drinking water
but fit for drinking works just fine.
Note that there isn't really any ambiguity in either of your examples, so the meaning is clear even if you use the wrong one. However, it is just about possible to contrive a use of this which would be misunderstood if used wrongly:
The suit is fit for a king
The suit fits the king
The former means that the suit is of very fine quality, good enough to be worn by royalty, but says nothing about its size. The latter means that the suit is the correct size and cut for a particular head of a royal household but says nothing about its quality.
Supplimentary Question: Is it ok to say "this piece of puzzle fits that piece of puzzle"
To my ear (I am not an expert on the subtle rules of language, just a UK native speaker who is happy to help where I can) this is fine. Puzzle pieces are actually somewhat interesting, because the fit is two way.
In the other examples, the relationship is somewhat one way. A lock is made, with keys which fit that lock. You wouldn't say "the lock fits the key". Clothes fit a person, you wouldn't say "He fits his clothes".
But the puzzle pieces are equivalent. Because of this I would be more likely to say
"These two pieces fit together"
"this puzzle piece fits with that one"
also notice the use of the word "one" to avoid saying "puzzle piece" twice in quick succession. The way you wrote it isn't wrong, but I would be more likely to use one of these.