To obliterate something is to destroy it completely, leaving nothing behind.
To eliminate something is far less dramatic; it means to remove it from further consideration. For example, in a knockout tournament, a losing team is eliminated.
A classic example of how 'eliminate' is much less dramatic than 'obliterate' is Sherlock Holmes's quote:
How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?
You could not put 'obliterated' into that sentence and still have it read sensibly.