How did Indian chess compare to the version that Europeans brought back to India?

The Mughals had a bit of an addiction to chess, it seems. One Mughal emperor (Akbar, r. 1556-1605) played Harry Potter-style "living chess" in the palace by making his servants pretend to be the pieces. A famous seventeenth-century match between two players in the service of the Mughal and Persian emperor respectively lasted for three days, and an Englishman reports that it was customary for hosts to amuse their guests with a chess match after dinner. Niccolao Manucci, an Italian traveler who spent decades in Mughal Emperor Aurangzeb (r. 1658-1707)'s court, says:

All the great men are fond of amusing themselves with chess-playing by which, as they say, they learn to govern, place and displace, give and take, with discretion, to the glory and gain of their projects.

Chess began in India, of course, and the variety to be found in the Early Modern Subcontinent was immense. The Mughal emperors themselves probably preferred shatranj, the Muslim variant of chess that is the direct ancestor of modern Western chess. The setup of shatranj is identical to Western chess, and the only major differences are that the queen (called a vizier in shatranj) only moves one space diagonally, the bishop (called an elephant in shatranj) can only move exactly two spaces diagonally (e.g. from A1 it can only move to C3), and pawns can't move two spaces even on their initial rank. There were some slight Indian touches on the rules of shatranj under Mughal rule; in Indian games of shatranj, it was often common to allow the Queen/Vizier to jump over the pawn in front of her (i.e. Qd3 even with a pawn in d2), or to allow the first-moved pawn to advance two spaces.

Compared to Western chess, shatranj is a much slower-paced game that emphasizes strategy more than tactics. The Queen, the Bishop, and the Rook can all attack from a distance in Western chess, but only the Rook can in shatranj.

European observers also claim that Mughal nobles played a four-player chess involving dice, which is almost certainly the chess-like game known as chaturaji. In chaturaji, a die roll decides which piece can be moved, the piece that moves the same way as a shatranj Bishop/Elephant is called a Boat, this Boat can sometimes capture three pieces with a single move, and the goal is to collect predetermined points and not checkmate. The Wikipedia article is worth checking out.

There were many other variants of chess under Mughal rule. The early seventeenth-century encyclopedia Bhagavantabhaskara introduces the rules for yet another type of chess. Some of the differences between this "Intellectual Game" (krida buddhibalashrita) and the chess familiar to us include:

Each Rook had one opportunity to jump over another piece, but they couldn't capture this way. This was presumably a way to speed up the game by allowing these powerful pieces to join the actor as quickly as possible by jumping over the pawns.
The chessboard had "X" marks on certain squares on the 1st/8th rank. If the pawn was promoted on an X-marked square, it stayed there; if it was promoted on a blank square, it went back to its original rank.
A stalemated player had the right to move(!) one of the enemy pieces preventing him from moving his King, and send his King to the resulting new safe square.

An early eighteenth-century book of chess, the Chaturangavinoda, introduces yet another variation of the game:

A piece called a Chariot begins where the Rook should be in chess and shatranj (a1 and a8, h1 and h8), but it moves like a shatranj Elephant (exactly two squares diagonally).
A piece called an Elephant begins where the Bishop/Elephant is in chess and shatranj (c1 and c8, g1 and g8), but it moves like a Rook.
The Queen/Vizier moves all along the diagonals like a bishop does in chess, not just to the adjacent diagonal space like the shatranj Vizier.

There were yet stranger variants of chess in Early Modern India. These included:

Maddat mari: An extremely slow-paced version of chess that could last for days, where no piece that was protected by another piece could be captured! The only way to capture a protected piece was if the capture creates a discovered check on the enemy King. The King could also move only when in check.
Sarvatobhadra: White has all the pieces, while Black has a single King called the sarvatobhadra (Sanskrit for "Auspicious on all sides"). Here's the catch: the sarvatobhadra moves like a Rook, a Knight, a shatranj Vizier, and a shatranj Elephant put together. As mentioned, shatranj and other Indian types of chess did not have fast-moving pieces like the Western Bishop or Queen that could attack from afar other than the Rook, so this was actually surprisingly balanced.
(There's a modern Indian variant of this game called Maharajah and the Sepoys, where the White pieces move as in modern Western chess and the Black King moves like a Queen and a Knight put together. This game is more imbalanced than sarvatobhadra because modern chess pieces, and hence White, are far stronger than shatranj pieces. Actually, if both sides played Maharajah and the Sepoys perfectly, White would win every game.)
Harihara Chaturanga: Invented by a minister named Godavaramishra in the early fifteenth century, this chess variant is said to symbolize the conflict between order and transformation and features 64 different pieces in a 196-square chessboard. The rules of Harihara Chaturanga are extremely complicated, but to sum up a few points:
Each side has thirty-two pieces of ten different types: one King, one Prince, one General, one Minister, four Horses, four Elephants, four Chariots, four Machinists, four Bowmen, four Spearmen, and four Swordsmen.
The King moves like the Western queen. The Prince, the General, and the Minister also move like the Western Queen, but only up to six squares away. The Chariot moves like the Rook. The Horse can move like a Knight, or it can move six squares like a Rook, or it can move like a Knight from a square six squares orthogonally away. So a Horse in a1 can move to b3, a7, or even b9. Movement of the Horse from the center of the board.
The Elephant moves one to five squares forward, or one square diagonally backward from its starting square or any of the five squares in front of it. I'm not explaining this well, but an Elephant in b2 could move to any square between a1 and a6, b3 and b7, and c1 and c6. Movement of the Elephant from h5.
As for the Swordsman, Spearman, Bowman, and Machinist... Actually, I won't bother explaining myself. Here's a picture of how they move.
There is no fixed initial position. Godavaramishra suggests a few optimal initial positions with names like "Fish muzzle position," "Chariot position," "Circular position," "Half-moon position," "Invincible position," "Thunderbolt position," "Crab square position," "Coiled snake position," "Firm ground position," and a simple "Strong position." Here's what "Fish Muzzle" vs "Half-Moon" looks like.
For obvious reasons given a King that can move like a queen, there is no checkmate and the goal is to simply capture the enemy king.

Because it was impossible for any normal human being to master, and I'm only half-joking here, Harihara Chaturanga never really caught on.

A good source for the global variety of chess variants is A World of Chess: Its Development and Variations through Centuries and Civilizations by Cazaux and Knowlton.