OK, I have lived and studied in Tokyo for 4 years, I play chess and I learned to play shogi over there. I am a better player of chess than of shogi.
My personal opinion: Both games are more or less equally good, and both have strategic depths, but those depths are located in different “oceans”, so to speak.
Western chess is, compared to shogi, a very fast game: Plenty of pieces (the bishop, the rook, the queen) are very far-ranging and tremendously powerful. The pawns in western chess are much more flexible than the pawns in other types of chess (western pawns can move two squares when moving for the first time, and capture in a positively funky way by moving diagonally). All of that makes for a game that requires the player to keep in mind every square in the board, to watch for threats that might potentially come from anywhere, and with a lot of potential and variety in the moves.
Shogi is a much more “sedate” game – the pieces tend to be short-range, there is only one rook-equivalent and one bishop-equivalent per side, and quite a few pieces (the knight-equivalent, and something called the “lance”) can only move forwards. Also, all pieces, in shogi, can be promoted, not just pawns (and not just by arriving to the last row, but on reaching any of the last 3 rows). However, promotions are fixed (each piece promotes to something fixed beforehand, you cannot choose what you would want to promote your piece to.
In my opinion, this makes for a game where you have to concentrate your strategy into slowly building an offensive at a much more sedate pace than western chess. If western chess can be lightning, shogi is like a glacier relentlessly advancing.
There is also an aspect of shogi that profoundly affects gameplay: Captured pieces can be “parachuted” anywhere on the board (there are some restrictions), becoming pieces for the player who captured them (shogi pieces are not distinguished by colour – They have all the same shape reminiscent of an arrow, and they point towards the enemy. Pieces have their names printed on them). If a player “parachutes” a piece on the board, he or she forfeits their move in that turn.
You have to keep track of the pieces that you have lost to your enemy, and must keep in mind that one of them might appear, literally, out of the blue to utterly wreck your position.
To sum up: Both games are strategically deep… It is just that the type of strategic thinking needed for each one is different.