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Is there a difference between a "plane of existence" and an "extradimensional" place?

In particular: can the telepathy spell (PHB, p. 281) connect two characters, one inside the extradimensional dwelling created by the Mordenkainen's magnificient mansion spell and the other in the same plane as the portal to the magnificent mansion?

Yes, there is a difference. They are not the same plane of existence and the Telepathy spell will not work in the situation you have outlined.

The Dungeon Masters Guide (DMG) has this to say on Demiplanes:

Demiplanes are extradimensional spaces that come into being by a variety of means and boast their own physical laws. Some are created by spells. Others exist naturally, as folds of reality pinched off from the rest of the multiverse. Theoretically, a plane shift spell can carry travelers to a demiplane, but the proper frequency required for the tuning fork would be extremely hard to acquire. The gate spell is more reliable, assuming the caster knows of the demiplane.

A demiplane can be as small as a single chamber or large enough to contain an entire realm. For example, a Mordenkainen’s magnificent mansion spell creates a demiplane consisting of a foyer with multiple adjoining rooms, while the land of Barovia (in the Ravenloft setting) exists entirely within a demiplane under the sway of its vampire lord, Strahd von Zarovich. When a demiplane is connected to the Material Plane or some other plane, entering it can be as simple as stepping through a portal or passing through a wall of mist.

From this we can see that the space created by Mordenkainen's Magnificent Mansion (MMM) is a Demiplane.

The Telepathy spell has this text:

The creature can be anywhere on the same plane of existence as you. The spell ends if you or the target are no longer on the same plane.

From the DMG rules on demiplanes it is clear that the MMM demiplane is not the same plane of existence as the plane where the entrance to MMM was created from. This is made explicit by the fact that the Plane Shift spell requires a tuning fork tuned to the demiplane specifically to access them.

We can see this from Plane Shift's material component:

a forked, metal rod worth at least 250 gp, attuned to a particular plane of existence

As a result Telepathy will not work between outside MMM and inside MMM.

While there are no official rulings regarding Extradimensional and planes of existence in the Curse of Strahd Alterations to magic.(Placing it in spoiler as to not to spoil the game for people.

Magic that summons creatures or objects from other planes functions normally in Barovia, as does magic that involves an extradimensional space. Any spells cast within such an extradimensional space (such as that created by Mordenkainen's Magnificent Mansion ) are subject to the same restrictions as magic cast in Barovia. --Curse of Strahd P24 Alterations to Magic

From the fact that this is an alteration to the existing rules we can deduce that according to the original rule spells that are cast in the extradimensional spaces are not bound to the plane of existence the space originates from.

Considering that each plane has it's own traits and extradimensional space is seperate from the said traits we can say that extradimensional spaces are not part of the plane they originate from thus disallowing spells and abilities that require being in the same plane such as telepathy.