In the 1950's algebraic geometry was tedious and hard to grasp because it was mostly commutative algebra, developed by Zariski and Weil and their schools to fill logical gaps in the Italian arguments of the previous half century. The rich geometric texture of the italian school was lost. In the 1960's Serre and Grothendieck introduced homological algebra to the subject and greatly expanded and enhanced it to embrace also arithmetic, but the abstraction level went WAY up, so again it was hard to grasp and relate to geometry. Hartshorne is a member of both Zariski and Grothendieck's schools and appreciates down to earth objects like space curves, but his book has a long beginning section on schemes and cohomology that can definitely throw a beginner off the horse.
Pardon the delay in getting here, but the point is that Shafarevich's book has none of the tediousness of the previous generation, yet benefits from the rigorous foundations via commutative algebra of Zariski's works. I would say Shafarevich's book, is a geometrically oriented explanation of the material that can be explained using Zariski's methods. I.e., it has a rich geometric feel, is very well explained, includes many easy examples, and is rigorous in its use of commutative algebra. This book allowed many of us who were stymied by the huge amount of algebra needed for 1960's Grothendieck style AG, to finally gain admission to the subject. I love this book.
There are three themes one can mention in algebraic geometry, 1) projective varieties, 2) schemes 3) cohomology. The first topic concerns the objects most geometers are interested in. The second one is of more interest to number theorists, but also has value for geometers in understanding limits of varieties. The third is a powerful and abstract technical tool which is valuable to both schools.
So here is the difference between Shafarevich and Hartshorne: Shafarevich focuses mainly on varieties, as the title reveals, has nothing at all in volume I on schemes, and neither volume treats cohomology. Hartshorne quickly reviews varieties then goes straight to schemes and cohomology, presenting schemes before cohomology. Thus Shafarevich is much more elementary than Hartshorne, and is a good introduction to that book. For a book treating cohomology before schemes, try George Kempf's algebraic varieties, but hartshorne will eventually be essential.
A few negative aspects to Shafarevich: it has a large number of typos and outright errors, in spite of numerous editions and reprints. One of the worst mathematical mistakes is the false assertion in the section I.6.3 of the first edition, that for any regular map f:X-->Y the set of points of Y over which the fibers of f have dimension at least r, is closed in Y. One needs the map f to be closed, e.g. proper, for this to hold, and unfortunately this false result is used in the treatment of the tangent bundle of a variety. The proof of the normalization for curves seems also flawed to me. All these mistakes can be fixed, and even in the present flawed form, this book still has no peer in readability and geometric richness with the same scope (although the undergraduate book by Miles Reid, Shafarevich's translator, is excellent for what it covers). I recommend it highly.
I would give it 4 stars for the errors, to leave room for a higher score for a book not only well conceived and well written but also well edited, but there is no other book that approaches this one in its strengths so how can one give it only 4 stars? But how can this book go through so many printings and not repair the most egregious mistakes? (There are other famous books out there with more and worse errors, but we who do not write books should not be too harsh in this regard. What if no one had the courage to attempt to explain this difficult stuff to others?)