Does it matter how you cut garlic?

Yes, it does.

Garlic contains volatile oils that are released when it's crushed or pressed. As garlic's cell walls are smashed, its oils react with its natural enzymes, and the aroma and flavour become very strong. If used immediately in raw preparations, the pungent pulp and extracted juices from pressed or pureed garlic give your dish a pronounced spicy flavour. Unfortunately, these oils don't last but turn rancid quickly and linger on hands, breath, and cutting surfaces. Pressed garlic doesn't hold up well when heated, either. It turns bitter and quickly loses its characteristic garlic flavour. I would advise against using a garlic press for these reasons. Plus, they are difficult to clean.

When you mince or chop garlic, the oils aren't violently forced out but are left to slowly season your food as its cooks. Also, the enzymes that make garlic pungent are destroyed by heat, so the garlic flavor is more apt to mellow as it cooks.

In general, the longer the cooking time, the larger you can leave the pieces of garlic. Finely minced garlic may also be used in vinaigrettes and salsas where the high acidity of the food will help break down the garlic and bring out its flavours.

The more aggressively garlic is handled, the more aggressive and short-lived its flavour.