Use adjusted prices to compute returns, volatility and signals. Use raw prices to decide whether an order was executable at a given level, to model tick sizes, and to apply per-share cost structures.
The subtlety is that adjustment factors change over time. A price series downloaded today and the same series downloaded next year will differ for every date before the next dividend, because a new adjustment factor was applied to all history. Backtests are therefore not reproducible unless you store the snapshot you used.
Worked example: a stock closed at 40.00 the day before a 0.50 dividend. After adjustment the historical close becomes 40.00 x (1 - 0.50/40.00) = 39.50, so the ex-dividend drop no longer looks like a 1.25% loss.
Related: corporate-action-adjustment, point-in-time-data, data-vendor, simple-returns