It is usually considered an error in both British and American English (though it is acceptable in some other languages, such as French). It is condemned in The Elements of Style, a popular American English style guide by E.B. White and William Strunk, Jr.
Simply removing the comma does not correct the error, but results in a run-on sentence. There are several acceptable ways to correct this:
Strunk & White note that splices are sometimes acceptable when the clauses are short and alike in form, such as:
(Examples adapted from the online, public-domain 1918 edition of The Elements of Style.)
Some writers are less prescriptive. This is what Fowler (third edition, 1996) has to say: "We are all accustomed to the ... conjoined sentences that turn up from children or from our less literate friends... Curiously, this habit of writing comma-joined sentences is not uncommon in both older and present-day fiction. Modern examples: I have the bed still, it is in every way suitable for the old house where I live now (E. Jolley); Marcus ... was of course already quite a famous man, Ludens had even heard of him from friends at Cambridge (I. Murdoch)."
The comma splice is often considered acceptable in poetic writing. The editors of the Jerusalem Bible translate Isaiah 11:4 as:
Lynne Truss observes: "so many highly respected writers observe the splice comma that a rather unfair rule emerges on this one: only do it if you're famous." She cites Samuel Beckett, E. M. Forster, and Somerset Maugham. "Done knowingly by an established writer, the comma splice is effective, poetic, dashing. Done equally knowingly by people who are not published writers, it can look weak or presumptuous. Done ignorantly by ignorant people, it is awful."
Grammarians differ as to whether a comma splice also constitutes a run-on sentence. Some define run-on sentences to include comma splices, but others limit the term to strictly mean those in which independent clauses are joined without any punctuation, excluding comma splices.