Do obliterate and eliminate have the same meaning? They are spelled very differently.
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Obliterate has the connotation of total, complete, utter destruction of a thing. It is a very strong word. Though it can be used figuratively, e.g., one might say "I obliterated my opponent in a chess match," it has a very strong sense of connoting physical and tangible destruction. To obliterate an object is to utterly and irrevocably remove its identity; an obliterated object can no longer rightly be described in the same sense as it previously was, in marked contrast to an eliminated object. The punishment of damnatio memoraie, that is, the effacing of all traces of some unlucky individual's name and public accomplishments from official records, and which was thought by the Romans to be one of the most severe they could mete out, is often translated from Latin to English as being "obliterated", which gives yet another clue to the gravity of obliterate's connotation.
In contrast, to eliminate has a much more tempered connotation than obliterate. It can be a very mild word meaning simply "to remove," though as a result of Cold War spy-games it has also gained currency for use in a coldly euphemistic sense to mean "to kill someone" or to "dispatch someone." The word elimination finds use in the sciences as well; for example, one of the most fundamental things one learns in an Organic Chemistry course is the elimination reaction, an apt term for the process by a which a molecule ejects or removes its one of its former constituents, its leaving group. For a doctor who specializes in treating the alimentary system, to eliminate finds use as a polite synonym for more colorful words meaning "to poop"; asking "Do you have trouble eliminating waste?" doesn't provoke the same level of embarrassment from a hypothetical patient as a result of the term's clinicality and "removedness."
The dictionary definitions ought to make the distinction clear.
Here's my understanding of the terms
Obliterate - destroy utterly, pulverise into nothingness, remove all trace.
Eliminate - destroy or (importantly) remove elsewhere.
You can eliminate a bad smell by moving the kipper from under your sofa into the trashcan. There is no need to obliterate the fish (the process of obliteration might be counter-productive)
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:
You could not put 'obliterated' into that sentence and still have it read sensibly.