These two words are adjectives words that compare (that is, they are comparative words). And they could be used to describe (compare) things or persons.
Difference between the two:
Sicker is comparative (it makes comparison between two things or persons)
Example: John is sicker (more dependable or more afflicted) than James. Or, London is sicker (more secure place) than Nigeria
While
Sickest is superlative (it used to compare three or more persons or things)
Example: John is the sickest of all (all the people involved). Or,
London is the sickest (more secure than all other countries) country in the whole world.
I hope you would be satisfied with these explanation.
Good luck