Stockfish depth calculations I'm using Stockfish to analyze my blunder filled games for blunders.
Let's say you send stockfish a FEN for a position and then calculate to a given depth ...
position fen the-starting-position
go depth 15

Now let's say a move has been made. I can either send stockfish the move...
position move the-move

or send it the fen that results from the new move.
position fen the-postition-after-the-move

Is Stockfish more efficient when you send it to the former?  That is, does it remember the evaluation from the starting position and use it as a starting point for analyzing the new position after you give it a 'position move ...' command?  And does it not remember the prior calculations if you just send it the next FEN in the sequences?

And a similar question.  If you tell Stockfish to analyze to a depth on a position, and then tell it to analyze to a deeper depth, does it start that second analysis from where the prior analysis stopped?
position fen the-starting-position
go depth 15
go depth 20
UCI protocol
Based on your answer, it looks like there is a typo and a misunderstanding.
The typo: It should be position ... moves, not position ... move.
The misunderstanding: The UCI protocol is stateless in the sense that you need to give the entire list of moves leading to the intended position every time, or the position itself.
Examples:

position startpos moves d1b1 is invalid, because d1b1 is not a legal move in the starting position.

position fen 3rr1k1/1b3p1p/6p1/pp1p3B/3N1P2/P3P2P/1p4P1/1R3RK1 b - - 1 28 is valid

position startpos moves e2e3 d7d6 d1f3 b8d7 f1c4 e7e6 d2d4 d6d5 c4d3 g8f6 c2c4 f8b4 c1d2 b4d2 b1d2 d5c4 d2c4 e8g8 c4e5 d7e5 d4e5 f6d5 f3e4 g7g6 g1f3 c7c6 e1g1 b7b5 a1d1 a7a5 a2a3 c8a6 f3d4 c6c5 d4c6 d8c7 e4d5 e6d5 c6d4 c7e5 d4f3 e5h5 h2h3 a6b7 d3e2 c5c4 f3d4 a8d8 e2f3 f8e8 f3h5 c4c3 f2f4 c3b2 d1b1 is valid and sets up the same position

position startpos moves e2e3 followed by position startpos moves d7d6 is invalid, because d7d6 is not a legal move in the starting position.


Differences between fen and moves
It's up to the engine if it can do something useful with the list of moves. For Stockfish, it's almost the same as not having the list, with no difference in efficiency. But having the list of moves informs Stockfish about possible repetitions, so it can avoid or aim for repeating an earlier position three times.