Sometimes you can use both as "meter", in fact it is usual to say "ponlo dentro de" instead "mételo en..."
I'd say that you tend to use "meter" when the action implies that fisically one thing goes inside another as a kind of a part of it or fits in it in a real of figurative way.
Poner or (poner dentro), I'd say that it is more used when it is more obvious and it is not a relation of "fit in" as puting clothes in the wash machine or putting things into refrigerator...
But don't take this as a rule, it is a mere thought about a natural way I tend to use both in that sense.
For example I tend to say: Ponlo en la nevera but mételo en el cajón.