Have saved games been compatible after patches?

Yes. Factorio has historically allowed saved games to be migrated between versions, and the 0.14 to 0.15 migration is no exception. Factorio 0.15 will be able to load any save that was made in Factorio 0.12 or later -- from the patch notes:

• Disabled loading of saves before 0.12.0 version (You can use 0.12 to load older saves and re-save them).

Generally speaking:

• Terrain will be preserved. If the game's terrain generation changed significantly between versions, there might be discontinuities between old and new terrain, but I don't think there's any such changes in 0.15.

• Some ore patches will probably be created for the new uranium ore.

• Most structures and items will be preserved. Some buildings which changed sizes between versions (like the fluid pump) may need adjustments, and I'm not sure what will happen with items that have been removed in 0.15 (like alien artifacts and science).

• Any research you have completed will still be researched. You may need to research some new technologies if new prerequisites were added.

Note that while your version 0.14 savegame might be technically compatible, your version 0.14 factory will likely not be.

In version 0.15, many recipes were changed. Most notable are the completely new formulas for science packs beyond red and green, but there are also a lot of other, more subtle changes (armor-piercing ammo now has normal ammo as an ingredient, for example). So large parts of your factory will likely not work anymore.

That means that when you have a game you want to finish, you should keep doing that with version 0.14.23. When you are using Steam, you can downgrade by right-clicking on Factorio in your games library -> Properties -> Betas and selecting "0.14.23" from the dropdown. If you want to give version 15 a try, it is really recommended to start a new game.