They are only job titles and, as such, would vary a lot between different companies. But here are some rough ideas:
"trainee" implies that they are actually training to qualify as an accountant, and as been pointed out by Yaffle , will probably get study leave and education expenses.
"assistant accountant" would suggest that they are not qualified. As a qualified accountant, I don't think I would like the the word "assistant" in my title. An assistant accountant might be studying or might have given up.
"Accounts assistant" would suggest a tecnician level person who helps out with the accounts e.g. debtors, creditors, bank rec etc, but would probably not be doing general ledger work.
Brendan