Let's assume that for whatever strange -- probably magical -- reason, the Moon was suddenly and instantaneously replaced by a planet identical to Earth in terms of mass and size. Its orbital path is identical to that of the former Moon. Aside from deciding how the planet got there in the first place, and where the former Moon went, what would the effects of its appearance and presence be on us Earthlings? For some extra fun, let's also assume this other Earth-like world has its own biosphere, and surface conditions similar to Earth (atmospheric composition and pressure, surface water, soil composition, etc).
The initial effect will be catastrophic, as the two planets adjust to a new orbital equilibrium; elliptic orbits around their common center of gravity midway between the two planets (see figure). The semi-major axis will be equal to half the current distance between the Earth and the Moon, while the semi-minor axis will me far smaller (I can't off-hand figure out exactly how small). The orbital period will also be shorter than the current lunar month.
The tidal forces between the two planets will be very strong, particularly at closest approach. This is likely to also cause strong seismic activity, as seen on Jupiter's closest major moon, Io.
The planets will eventually be tidally locked, with the same sides facing each other. As rotational speed is bled off, the orbits will increase because the angular momentum of the planets is transferred to the angular momentum of the orbit. This will reduce the eccentricity of the orbit, since the effect is strongest at closest approach.
Once the planets are tidally locked, there will be no 'lunar' tides as we know them any more, as high tides always face directly away from and towards the orbiting body. The height of these tides will however vary with the distance between the planets, with the highest tides at closest approach (and correspondingly lower tides at the 'sides' of the planets). There will also still be solar tides, which account for almost one-third of tidal forces affecting the Earth.
An additional complication is that there will be some 'wobble' in the positions of the sister planets in the skies, since the planets rotate at constant speeds, while they pass each other at much higher angular velocities at closest approach than when they are farthest apart. This will create a pull or push on the planets' surfaces that will move tides from side to side during close approach.
If you magically replace the Moon with a copy of the Earth, two things happen:
1-) Both planets will immediately have their surfaces ravaged by giant tsunamis. The Earth has about 100x the mass of the Moon, so tides will be extremely strong.
2-) They will spiral towards each other and eventually collide. The collision would happen in less than four days. The reason is that there are forces acting on the Moon keeping its orbit almost in equilibrium (its orbit is expanding slowly). The force pulling the new planet towards Earth would be almost 100x stronger, so the new vector would mean catastrophe for us.
The last time the Earth collided with another planet, the debris formed the Moon. But it was a Mars-sized planet last time. This time, a new Super-Earth will be formed. It will start out in a Hadean phase, so it won't be inhabitable. With the heat of the impact all water will evaporate into space, and without the asteroids for a new bombardment, this new resulting planet may never develop a hydrosphere.