One of the first things that might explain why the collectibles in Tomb Raider were much better than those in Assassin's Creed 3 is that it makes more sense for Lara Croft to want to collect the items she comes across than it does for Connor to. The lead of Assassin's Creed 3 was focused on murdering Templar agents in pursuit of a better world. Because of this, it made little sense for him to spend his time running around the city to collect pages from Benjamin Franklin's Almanac's, scouring the forest in pursuit of eagle feathers, or going around breaking into chests located throughout the world. With the possible exception of the Peg-Leg trinkets (and even that is running on a pretty weak case in my honest opinion), all the collectibles do not really tie in to the character or the narrative. They seem completely superfluous and only exist for the sake of having them.
This is not true for Lara Croft in the Tomb Raider. Much of what the game lets players collect would make perfect sense for Lara to take the time to gather. Since she is in the middle of a survival situation, it would make sense for her to want to gather salvage and weapon parts from the area. Another easy to explain collectible are all of the various artifacts Lara can find. While some might say that it makes little sense to want to look for these kinds of artifacts in a survival situation, Lara is an archeologist by trade, inherently interested in objects that detail the kind of experiences and lives people on the island would have led. Even in the story, there are several times during the course of the plot where Lara takes a break in order to examine whatever tomb or vestige of the past that she currently occupies. This curiosity of her's extends to collecting the journals left behind by the many inhabitants of the island, both past and present, that players can gather. As someone who studies people and culture, documents of any kind are of intrinsic value to Lara. While it may not be necessarily true for all collectibles, in the general case Lara Croft would have a good reason to collect most of the optional items in the Tomb Raider reboot.
The second reason Tomb Raider has, in my opinion, better collectibles is that players have a better external, in-game motivation to collect them. In Assassin's Creed 3, a lot of the things players could collect did not have much worth in terms of rewards for collecting them. Gathering up all of the eagle feathers scattered throughout the Frontier merely unlocked a Native American tribal garb that visually was not very impressive or distinctive in any way. Another solid example of a lack of rewards are the Almanacs. When Connor finishes collecting a set of Almanac's, they unlock a blueprint for one on Benjamin Franklin's inventions which can be crafted at Connor's Homestead. While this seems like a big deal, the inventions are static objects with no other purpose but to decorate the Homestead, conferring no practical benefits to players. Lastly, the treasure chests do offer material rewards, but I can hardly call them particularly beneficial. What I mean is that chests contain materials, recipes, and money which can all be used in different ways to take advantage of the in-game economy. As I have mentioned in my critique of Assassin's Creed 3, the in-game economy has no real use to players and can be skipped entirely without suffering for it or losing anything that one might get by participating in it. If the chests' rewards feed into a worthless add-on, it can be said that they are also worthless add-ons.
When contrasted that with Tomb Raider, this lack of rewards becomes even more glaringly obvious. Everything that can be collected in the game confers at least some sort of reward. Performing any task, even collecting an object, gives the player XP which are used to get skill points that can be spent to unlock new talents and passive skills for Lara, giving players an extra edge against enemies and the environment. In this way, Tomb Raider already offers more practical rewards than Assassin's Creed 3 ever did, but that is not the only way they reward players. One of the most commonplace and ubiquitous collectibles in the game is “Salvage.” Salvage is like XP in the sense that players can spend salvage to upgrade the weapons Lara finds over the course of her adventure. Each weapon can be upgraded with a limited selection of upgrades. This is where another, slightly less prominent collectible comes in: The weapon parts. Weapon parts come in different categories: bow, pistol, rifle, and shotgun. When the player collects enough parts in a given category, Lara will upgrade the corresponding weapon to a stronger version of itself (with all current upgrades transferring over) when the player reaches a campfire. Upgrading a weapon improves its abilities and unlocks more upgrades which players can use Salvage to unlock. Tomb Raider actively rewards those who take the time to explore and hunt for these collectibles. This makes the process far more enjoyable, even if it is partly due to a Skinner Box-like effect.
This is not the only way in which players get rewarded by what they do. Games can offer more for collecting than simple gameplay benefits. Another way in which Tomb Raider has better collectibles is that collectibles can also further develop the characters and setting of the game. When Lara finds an artifact and picks it up, she gives players a brief description of what it is and what significance it has. These artifacts can tell us a little about past civilizations and people that have landed on the island and lived there for long stretches of time. It helps players understand exactly how many people have been impacted by the island and its many secrets, working as a form of world building. Also, as an archeologist, Lara does not just magically know everything about a given artifact. There are many of them in the game that players can rotate around to a specific angle to learn additional information. One example in particular stands out to me: In one area, Lara can find a small dragon statue, making the remark that it looks like a priceless antique. If one rotates it so that the bottom faces the camera, she will say “Oh! It's a fake. 'Made in China'”. Other examples just help to bring players into the world and the role of Lara Croft.
As I said earlier, the characters are also developed through collectibles, not just the world. This comes in the form of all the logs and journals players can track down. Whenever the player grabs a journal, the character who wrote it narrates the text for the player. This can best be equated to the audio logs made famous by games such as System Shock 2 and its spiritual successor, Bioshock. There is a reason why this often works: It allows the player to learn more about a character and what makes them tick without needing to write up an entire conversation and animate it. Without spoiling anything, these logs make the characters more relatable and understandable. Combined with the artifacts, they also enable much more world building than either could alone. By collecting these items, the island and the people on it feel like a fully fleshed out world, and there lies the beauty in the way Tomb Raider handled them. I would try to find a comparable example from Assassin's Creed 3, but I do not believe one such example exists.
Lastly, the final reason I liked Tomb Raider's collectibles a lot more is because they are generally more challenging to get. Let me preface this by saying that in both games, all the collectibles are comparatively easy to find. Even if players have difficulty doing so normally, there exists ways players can make discovering the location of collectibles easier: Assassin's Creed 3 has maps which Connor can purchase to point out the location of every collectible in the game, while Tomb Raider allows Lara to upgrade her “Survival Instincts” (similar to Batman's Detective Mode in Arkham City), to highlight them and place them on the map when they are in range. (For the record, that is a good thing.) However, once the collectibles are located, getting to them is different in both games. In Assassin's Creed, it is typically pretty easy. Once Connor is roughly aware of a collectible's location on the map, it is a trivial task for him to get to it with the usual set to traversal mechanics. This is not the case in Tomb Raider. While some are certainly easy to get, many of them will take a degree of problem solving to reach. At other times, it is not even possible to collect a given treasure until a certain piece of Gear has been acquired allowing Lara to bypass an obstacle in her way, like classic Metroidvannia-style games. Having to actually bypass an obstacle by oneself to get a treasure, without the game doing excess hand-holding or making it too easy, gives a feeling of satisfaction that is otherwise absent, making the act of picking up a collectible all the sweeter.
In the end, Tomb Raider managed to do the impossible: It caused me to rethink my stance on collectibles in video games and give them the illustrious title of “Not inherently bad.” The only thing that has been able to remove itself from my shit-list in such a way is the Quick-Time Event. I am glad this game caused me to reevaluate my thinking, because for a long time collectibles were one of the fastest ways games could irritate me. It all started with those abominable feathers in Assassin's Creed 2 which were so annoying to gather up and for such an insignificant payoff. Nonetheless, collectibles are just another tool in a designer's arsenal to use and balance wisely. Like any tool, there are times where it works and times where it does not. A designer's responsibility is to know the difference.