Your question is a bit unclear, but an obvious difference between these two topoi is that the Recursive Topos is a topos of sheafs, hence cocomplete, wherease the Effective topos only has finite (non-trivial) coproducts. For example, the natural numbers object in the Effective topos is not a countable coproduct of 1's.

If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos models computability a la Banach-Mazur (a map is computable if it takes computable sequences to computable sequences) and the Effective topos models computability a la Kleene (a map is computable if it is realized by a Turing machine). In many respects Kleene's notion of computability is better, but you'll have to ask another question to find out why :-)

Your question is a bit unclear, but an obvious difference between these two toposes is that the Recursive Topos is a topos of sheaves, hence cocomplete, wherease the Effective topos only has finite (non-trivial) coproducts. For example, the natural numbers object in the Effective topos is not a countable coproduct of 1's.

If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos models computability a la Banach-Mazur (a map is computable if it takes computable sequences to computable sequences) and the Effective topos models computability a la Kleene (a map is computable if it is realized by a Turing machine). In many respects Kleene's notion of computability is better, but you'll have to ask another question to find out why :-)