Yep goo manager is what you use to update twrp, if it's uploaded for your device, it will appear when you choose to flash zips.
Rom manager is only for cwm recovery. It will only update cwm, and can only download cwm. Pretty biased actually, considering it's supposed to be a "rom toolbox".
Sprint GS3 Running TN's Msg and Chubbs
It can be updated using rom installer. Get it from playstore and rom installer > flash recovery> install latest version of your recovery
I have a root question. Where do i got to update TWRP?
Is there a reason you use flash if over twrp manager? Just curious.Use flashify app from Google store. After install click on recovery image...select download twrp...it will show all the latest twrp..select anyone...click yup