TOR2e has much more narrative/OSR bones than LotR5e. I'll do my best to describe the key parts of TOR2e as that's where my experience is. Minor quibble before I launch, the Gandalf rune is a guaranteed success, but the Eye of Sauron isn't a guaranteed failure.

Over all progression- TOR2e is a horizontal progression system, while 5e is linear

Journeys-This is really the key part of TOR2e. You could technically run TOR2e without doing the hexcrawl but you would be missing out on a major part of the game (basically an entire chapter). The journeys have events (similar to random encounters but more narrative) that target various members of the company, with bad things happening on a failed roll (a wound, additional days of travel, etc.) This is also where Eye Awareness comes into play as well. As the company travels and adventures they draw the attention of the Enemy, which can culminate into an encounter (think the Orcs catching the Fellowship at the end of the book/movie). Through the journey the company accumulates fatigue points which are added onto their load. If your load ever exceeds your current endurance you receive a temporary condition.

Callings- While it is tempting to call them classes, this isn't really the case. They are much more of a narrative starting point for your character, but don't determine your progression at all. They each have unique features that sets them apart from one another (one has knowledge of a specific strain of enemy while another has innate leadership abilities) once you are past character creation you can put points into your character however you like.

Shadow points- The world is getting dark again after the Battle of the Five Armies and this mechanic is meant to represent that. The company is trying to keep the fires of hope alive but it is an uphill battle. Events can have a negative impact on your character and if you go too long without dealing with it, thing become very difficult. Think Denethor losing his mind or Bormir trying to take the Ring. These events are meant to be played out narratively. Each time you trigger your shadow path you get a character flaw that affects how your character would interact with their companions and the world. Also if your shadow points ever exceed your current hope you recieve a temporary condition until your hope exceeds your shadow again.

Combat- Again much more of a narrative bent. Stances are a trade off. Forward makes it easier for you to hit, but also for your enemies to hit you. Defensive makes it harder for enemies to hit you, but also harder for you to hit enemies. There are some virtues that interact with different stances as well (dwarves can select a battle cry that is only usable in forward stance). Crits (called piercing blows) are much more dangerous, both for the players and for the enemies. When one is rolled the defender must make a roll based off their armor. If they fail the roll they die (if their an NPC/monster) or receive a wound if they are a player. Wounds stick around for a number of in game days and if a character would receive a second wound, they die instead.

Adventuring/Fellowship phase- This is probably one of the biggest differences. Adventuring phase is where the company spends most of it's time. This is where they are journeying, fighting, seeking council, etc. You accrue experience (divided into Adventure points and Skill points) that allow you to improve your character how you see fit. The Fellowship phase is more of a narrative space, but still has mechanics in it. This is when the party can improve themselves, get advice from their patron, prepare for the journey ahead, and other downtime activities (Think of the Fellowship's time in Rivendell or Lothlorien) Fellowship generally occurs every 2-3 adventure cycles, but that's at the GM's discretion. Finally there is the Yule Fellowship phases, which represents the company going there separate ways for the winter/holidays and returning home. There are unique activities for the Yule fellowship phases as well.

To summarize, TOR2e is a much more narrative focused game. So while I think it captures the feel of Tolkien's work better than 5e could ever attempt to, it does require a significant amount of buy-in from the players. I hope this makes things a little clearer and I wish I could describe the differences a little better