There is a subtle difference (which English doesn’t have because it uses the same word - recommended - for both occasions).
Cymeradwyo is more in the sense of approval - recommending a person or thing (such as a hotel for instance) by saying they’re acceptable.
Argymell is more in the sense of proposal - recommending something by offering advice.
I hope that makes sense!
It does! Thank you. I’ll file it in the box with gwybod/nabod - verbs that are distinct in Welsh, but not in English.
It might give a clue if I quote from Anthem Scarlets “Sosban fach yn berwi ar y tân” 
Adding to @siaronjames 's answer the word cymeradwyo can also mean applaud as in “Cymeradwyodd y dorf yn uchel …” (The crowd applauded loudly).
…well applause is approval after all! 
(the crowd approved loudly! 
)
Of course, I never thought of it like that. So obvious when it’s pointed out.
Ydy “hir yn byw’r Gymraeg!” yn gywir? for Long live the welsh language! ??
I’ve heard "hir oes i … " so “Hir oes i’r Gymraeg!” maybe.
Ahhh diolch. Dw i wedi trio dweud rhywbeth like “Vive La France!” yn Gymraeg! 
“Bydded i’r hen iaith barhau!”
Long may the old language continue.
As per the national anthem!
Cymru am byth!
Yes, traditionally a war cry I believe! …just sporting occasions/ competitive situations now!
Rich 
How to say "I have got ( a cat, a dog etc) in chatty North Welsh. I revert back to school learned “Mae gennyf fi” and I know it’s not that. Tell me and I’ll practice the modern equivalent all day and all night till it rolls off the tongue as though I’d never been to school
Because I’m in Australia I can’t just pop out and ask the neighbours 
Diolch yn fawr
The spoken form is actually quite close – Mae gen i, and some people say Mae genna i, which is even closer to the “school form”
Diolch Hendrik
If you fancy a chat in South Welsh it’s mae (gy)da fi …
“Kept safely”
I’m not sure how to say that something is “kept safely”. To keep is “cadw”, right? And safely is “yn diogel”? But I’m confused because in English “kept” is the past participle of “to keep”, so I’m not sure how to phrase something like “The phone is kept safely in a box”.
I think the most usual spoken form would be
Mae’r ffôn yn cael ei gadw yn ddiogel mewn blwch/bocs
lit. the phone has its keeping, safe in a box.
