I’ve not played any of these games, but I do own most – perhaps all? – of the Heavy Gear RPG books and some of the Heavy Gear wargame books (possibly a mixture of Silhouette and Blitz as I bought them for the setting / fluff rather than the rules themselves).   My understanding is that the Blitz rules are a streamlined adaptation / evolution of Silhouette, so although they might not be tournament legal / directly compatible with each other, I’m sure you’d be able to understand books bought for one game even if you played the other.

I was never able to get into the Battletech setting, but I absolutely love the Heavy Gear one.   It’s a rich, interesting history with lots of opportunities for telling / setting fantastic stories.   I would highly recommend it.   It also seems to me that heavy gears make more sense from a science / ‘realism’ perspective than huge mechs or titans do.   i.e. being smaller, more nimble war machines they fit into a niche in a modern/futuristic setting better than giant building sized machines, however cool they look.

The ‘core’ Heavy Gear setting features two alliances as the two main protagonists: northern and southern groupings that are kind of reminiscent of NATO and the Warsaw Pact.   In the history of the setting they ally with each other when Earth invades, but after winning that war, decades of mistrust leads to them falling out with and going to war with each other.   The other two main playable factions in the main game are the remnants of Earth’s invasion forces (who have become integrated with a local ‘non-aligned’ Terra Novan faction) and the other major ‘non-aligned’ faction, Peace River.

As the storyline progresses Terra Nova links up with other planets in the network, which brings Earth forces back into the game, along with Earth’s allies drawn from the other colony planets, but I’m not sure how many of them are represented with miniatures in the current Blitz game.