Untermiete is the legal relationship; Zwischenmiete is the everyday word for the common case of it. If you rent from someone who is themselves a tenant, you are an Untermieter and they are your Hauptmieter — the main tenant, who remains on the hook to the landlord for the rent and the state of the flat.
The distinction matters when something goes wrong. Your contract is with the main tenant, not the owner, so the owner is not obliged to deal with you and in most cases will not. If the main tenant stops paying the rent, the landlord can terminate their lease, and yours falls with it — you have no independent claim to stay, however punctually you have paid.
That sounds worse than it usually is. The overwhelming majority of Berlin sublets are somebody's flat, let by somebody who wants it back in good order, and they go fine. But it is the reason to care whether the main tenant has their landlord's permission, and the reason not to hand over a large deposit to someone whose surname you do not know.
One practical note: German law is on your side about the main tenant's right to sublet at all. A tenant who asks their landlord for permission to sublet a room usually has a legitimate interest the landlord cannot simply refuse. It is when nobody asks that the arrangement is fragile.