dnd 5e – Does the Mordenkainen’s Magnificent Mansion spell’s mansion exist on a separate plane of existence?

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.