I picked them both up. The Green Ronin book is a bit meatier, weighing in at 112 pages, with a price tag of $20. The Badaxe Games book is only 62 pages and is roughly 2/3 the size of the GR book, but costs half the price. Both books include new feats and prestige classes. Neither of them has much fluff on dwarven societies or roleplaying dwarves. Dwarven splatbooks basically.
The Badaxe book has interesting new twists on crafting rules.
The Green Ronin book presents a bunch of new items, both mundane and magical. It also a section on alternate dwarf concepts, even though they are only lightly touched upon. It also has some new creatures (mostly templates) and some new spells, including new domains and a type of magic called Clanheart magic (basically it's a spell list accessible to all spellcasting classes, both divine and arcane).
As for the feats themselves, the feats in the Badaxe book seem to be more balanced, but lack a lot of originality. I love the "Cleave-Asunder" feat though. The Green Ronin feats tend to be a bit on the powerful side. They introduce three new kinds of feats: Bloodgift, Bonding, and Channeling. I really liked the feats that gave you bonuses when fighting next to someone else with the same feats (Back-to-Back, Tandem Fighting, Protected Archery, Protected Casting, Shield Wall, Lock Shields).
As for the prestige classes, Badaxe decided they would produce one prestige class per multi-class fighter combo. So there is a prestige class that suits fighter-rogues, fighter-wizards, fighter-paladins, fighter-rangers, fighter-sorcerors, etc. The prestige classes themselves seem pretty interesting, and they should be easy to plug into almost any campaign. I particularly liked the Battle-Rager's "Against the Odds" ability.
Green Ronin's prestige classes are more unique and have a lot more flavor. I already want to play a "Knight of the Black Forge". Most of the prestige classes might need a little more work to plug in to an existing campaign though, particularly since some of them require feats from the book.
"Heroes of High Favor: Dwarves" was the first Badaxe Games book I've bought as it is their first product. The ideas inside seem to be solid, if perhaps a little on the bland side. For a first effort, I give them a big thumbs up and will certainly check out any future products.
As for "Hammer and Helm", I'm a big Green Ronin junkie, so it's only natural that I like it. I think it's a wonderful start to their new Races of Renown series. Now if they'd just hurry up and do one for halflings... (but, no, they've got to do one for Drow first <grumble>).