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Pat is a trusted home cook.
Here's a quick guide for you:
Nancy is a trusted home cook.
And here's another one.
But if/and you are wondering whether you would prefer the French to the Greek or vice versa?
Or maybe you like them all?
Well, a side by side tasting would tell.
The Bon Appetit guide is good for flavor/texture details, but here's the basic run down technically:
Greek feta - PDO Greek Feta, especially the ones imported to North America, is almost always a ewe/goat milk blend. Ratio is typically 70/30 ewe being the majority - that is the minimum allowed under the PDO criteria (goats are also quite a bit more productive milk producers than sheep, and require less grazing land). Barrel-aging is the traditional method - basically the immature cheese is stacked up in birch wood barrels with layers of salt. The salt draws out the large quantities of water left in the cheese and creates its own brine. It ages in the barrel this way usually for a minimum of 4 months. Naturally - this method is much more expensive, so you'll pay a premium for Greek PDO barrel aged feta. This aging process, however, makes Greek feta much sharper/tangier and somewhat less creamy than feta produced by faster methods like brining in metal containers. However, it does vary by producer - the more ewe's milk used, the creamier the feta. Greek feta crumbles readily so its very good as a topping and it can be grated as well.
Bulgarian feta - Usually called 'white [brined] sirene' in Bulgaria, exports of this feta are typically made with 100% ewe's milk. This cheese is eaten much less mature than the PDO Greek counterpart, and as such is usually matured for 1-2 months in metal containers. Because of the high fat content of ewe's milk, Bulgarian feta is quite a bit softer and has a much creamier mouthfeel than Greek feta. It doesn't crumble quite as easily but its easily broken up with a butter knife and is great in composed salads and on meze platters. Bulgarian feta is salty but its not overpowering, less tangy than Greek feta. Incredibly popular in the Balkans, former Soviet Union, and Israel. In fact in Israel most people refer to all feta as "Bulgarian cheese" regardless of origin.
French feta - tends to be mildest flavored, and medium texture/firmness, between Bulgarian and Greek feta. It also doesn't last quite as long as the other two. Made typically in Southern france where there are many ewe dairies. Valbreso brand, from Lactalis/Président is the most common label you'll see in North America.
Bulgarian feta is typically my go-to feta if I can find it, especially when I have homegrown tomatoes - but Greek is the one you want for flavor in cooking.