Weblio gives both the meaning of role or part (duty is there too as a less used meaning), so I assume the difference is on the nuances. In addition, 役割 seems to be much more used by Japanese natives than 役回り. The few examples I could find of 役回り carried negative connotations, either regarding duties or when it came to acting, it was used to refer to playing a villainous character. Does 役回り always have a negative meaning?
役回り is not necessarily negative, but it's something passive. 明鏡国語辞典 defines 役回り as:
(※回り合わせ = twist of fortune)
That is, you don't actively take some 役回り, but you passively (or sometimes randomly) end up being assigned to some 役回り. No one wants to actively take a bad role, so it is true that 役回り tends to refer to negative roles. But 役回り can also refer to a good role assigned to someone, as you can see in the following examples in BCCWJ.