Easy : a écriveur is what is colloquially called a writer of second class. Who writes a lot of letters, who loves to write but it is not his job
Beyond the different meanings, a big difference is for écrivain to be a common and widely used word while écriveur is, despite beying easily understandable, extremely rare.
There are two major differences, type of writer and obsolescence.
Écrivain (fem. écrivaine) designates a professional writer or someone whose main occupation consists in writing. The word is still in use.
Écriveur (fem. écriveuse) is obsolete, and is not found in modern dictionaries. It is not in the TLF, not in the wiktionnaire, not in Le Petit Robert (1994 edition), not the Grand Dictionnaire Encyclopédique Larousse, not in Le Dictionnaire Historique de La Langue Française, not in the Dictionnaire Culturel en Langue Française, I have found it only in Le Littré et le Dictionnaire Critique de la Langue Française de Jean-François Féraud, both dictionaries are considered as historical monuments but entirely obsolete.
Écriveur was someone who liked writing or wrote a lot, without a particular reference to the quality of their writing.
Féraud gives an example from Mme de Sévigné :
… ce n’est point un écriveux…
(Mme de Sévigné’s spelling). It means the person she’s talking about doesn’t like writing and doesn’t write much.
Dictionaries will usually mention three other words in relation to writers, écrivassier, écrivaillon and écrivailleur, all three are derogatory terms for second-class writers. They’re all used but I expect écrivaillon is the most commonly used.