Question

What is the difference between a Strong Style move and an Agile Style move?

Answer 1

Using a move in the Agile Style raises the user's action speed, which may make the users next turn come sooner, allowing for consecutive attacks. On the other hand, using a move in the Strong style will delay your Pokemon's next turn.

Source

If the move affects stats, Agile Style will only last for the user's next turn, while Strong will last for the user's next 5 turns (standard is 3). Finally, using either style will cost 2 PP instead of 1.

Comments

To add to this...  Agile Style reduces the power of the move used by 15 BP, while Strong Style increases it by 15 BP.  If the move has a chance for a secondary effect, that chance is reduced for Agile but increased for Strong.  If the move affects stats, Agile Style will only last for the user's next turn, while Strong will last for the user's next 5 turns (standard is 3).  Finally, using either style will cost 2 PP instead of 1.

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15 is straight up inaccurate. I’ve had boosts up to 30 and as low as 10.

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Even if the 15 BP was inaccurate (I don't believe it is, from my experience playing, but I admit I wasn't heavily scrutinizing it) everything else I said is true and can be tested in game.  The original answer was itself laughably bare-bones.

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I don’t think there’s a set number or percentage, it’s different for each move.

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Doesn't it affect Accuracy if the move's accuracy is less than 100?

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I was going to come back and edit the 15 BP to 20 BP, since I experienced this myself in game. The rest of PhailRaptor's comment I can confirm is correct. I will edit in the correct information.

Nought the Curspeon, styles do not impact accuracy.

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Strong style increases accuracy. I’ve gotten BP boosts of 10, 20, and 30 and drops of 10-20 BP, depends on the move.

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Can confirm that strong style impacts accuracy