First edition book worth more?

There are many variables involved.  

What is the title of the book?  If it was a Sherlock Holmes book, the first edition may be less valuable than "The Refugees" (a historical novel Doyle wrote) because it's likely that more copies of the Holmes book were printed and sold and so more copies are likely available. Or the Holmes book may be more valuable than "The Refugees" because Sherlock Holmes is famous and therefore a first edition of "his" book would be "in demand".  A first edition of "A Study in Scarlet" is usually more valuable that one of "His Last Bow" because "Study" was the first Sherlock Holmes title published, and "His Last Bow" was the second to last Holmes book.  There are also die-hard Brigadier Gerard and Professor Challenger fans that would pay more for a book about their hero than for a Sherlock Holmes book, though I suppose the demand for Holmes would shoot up the auction price of a first edition Holmes above the few bids for a Gerard or a Challenger book

Was the U.S. book published before the U.K edition or after it?  It might not make much difference except to a fussy Sherlockian collector, but Collier's Magazine bought the rights to publish the stories in "The Return of Sherlock Holmes" in magazine format a month or two before they were published in "Strand Magazine".  That meant that the Americans read about Sherlock Holmes' reappearance from the dead before the British did.  I don't know if the U.S. publisher of "The Return" had made a similar deal for the first rights.  There would be less than a year's difference in the publication date.

The condition of the book is also important.  A first edition book in fine condition would be more valuable than one in poor condition.

If the U.K. edition in your possession has a book-jacket (if that's what you mean by "drawings" on the cover), that jacket will add significantly to the value. Existent book-jackets on old hard-covered books are rare, especially if they are in good condition. Some of the illustrators were also famous enough to add significantly to the auction value.

  Some of the cover art can be a bit strange.  The Toronto Reference Library's Arthur Conan Doyle Collection has a Sherlock Holmes volume that has lilac blossoms on a white cover. (I'm not there right now and can't recall what the title is or who the publisher was. It was either "A Study in Scarlet", "The Sign of Four" or "The Adventures of Sherlock Holmes") I suppose the printer had a few hundred extra covers of a romance novel and that particular publisher (probably a "pirate": a publisher who sold books he did not buy the copyright to print.) thought "Why not? It's a cheap deal for me." (The Toronto Library bought the collection of Nathan Bengis, a collector of pirated copies of "The Sign of [the] Four")

Arthur Conan Doyle usually signed his letters "A.C.D." or "A. C. Doyle" (informally to his publisher's editor, his agent and his friends) and "A. Conan Doyle" otherwise. (The Toronto Reference Library has some of his letters. Those are the ones I saw. The collection at the University of Minnesota has other letters. The Portsmouth, England Public Library's collection has others, I'm sure.) The "Crowborough" edition of his collected works is not the first-edition of each of his works, but it is the U.K. edition (I think it was published as a tribute to Doyle after his death in 1930) Each book cover was impressed with his signature "A. Conan Doyle" on the lower right hand corner. It is a handsome looking set.

The one which is actually the first edition. Since the US version is a "first edition" in the USA not the World.