#Note: I'll compare both the Olympus 3.5's and 4.0's to the Lone Peak 4.5's in this post. However, I also laid out the differences between Olympus 3.5 and 4.0 as well as the differences between Lone Peak 4.0 and 4.5, so check them out if you're interested.

Lone Peak 4.5's in action.
The most important difference between Olympus and Lone Peak is in their cushioning (as it's the case with almost all Altra trail running shoes).
In summary, with everything else being average; if you're mostly going to be backpacking, trail running, walking and/or standing uninterruptedly, for long periods of time; then you're highly likely to benefit a lot from the extra cushion Olympus has over the Lone Peak's.
In Altra's trail running line, Lone Peak's are the closest you get to a typical trail running shoe and hence a safer choice than the Olympus and is probably a better option for beginner folks.
Additionally, Lone Peak 4.5 looks quite pretty (this Gray/Orange color of Lone Peak 4.5 is really liked by many - including me).
If you think you're in somewhere between these two points, then checking out my Timp 2.0 review as well as my comparison of Timp's to Lone Peak's can be a good idea.
In fact, comparing one of these trail running shoes to Timp's, rather than to each other, would make more sense, but never mind...
Cushion
While it's true that more cushion generally results in more comfort as either the distance to cover gets longer, or the individual gets heavier, or the trail gets rockier etc...
It's a myth that more cushion generally results in less damaging to our joints.
I wanted to make this clear before we get to compare these two products because cushioning would be my single most important criteria when deciding between them - although joint health shouldn't be counted as a criteria.
So, with everything else being average; if you're mostly going to be backpacking, trail running, walking and/or standing uninterruptedly, for long periods of time; then you're highly likely to benefit a lot from the extra cushion the Olympus possesses over the Lone Peak's.

Olympus 4.0. On the shelves around summer 2020.
If you're to be participating ultramarathons with these shoes, then generally speaking, I'd probably draw the maximum event distance line for Lone Peak's at 50 miles (80 km's). Anything longer and you're likely to wish having more cushioned kicks such as the Timp or Olympus.
This is NOT to say that Lone Peak's are an objectively better choice when you're below that limit. Because, if you're a relatively heavy person, more than, say, 200 lbs (90 kg); or the trails you'll cover will mostly be fairly rockier than average; or you're fond of having more underfoot protection for any reason etc...
...then a more cushioned alternative such as the Timp or Olympus can still be a reasonable choice.
BUT...
One strong downside of ample cushion is weaker toe-off when running - which is kind of important when running longer distances, and definitely important when running shorter distances.

Olympus 3.5's.
If you're unaware, here's what toe-off means: Pushing the ground farther from yourself not only using your entire forefoot but also your toe strength. Yes, this results in minuscule differences, but it adds up when you're competing with milliseconds (hint: Shorter distances).
This is the biggest downside of extra cushion.
After reading the entire thing until here, if you think you're somewhere between these two pairs, then I'd probably recommend you to check out the Timp 2.0's. It's probably what you're looking for.
This is it for the cushion part.
Now...
I'd even go that far and recommend you to downright IGNORE EVERYTHING else such as traction, fit, breathability, durability etc...
Not because they aren't important (they are), but because they don't differ much in our case - especially when compared to how cushioning differs.
Men's Lone Peak 4.5 & Olympus 3.5:
No products found.
Women's Lone Peak 4.5 & Olympus 3.5:
That said, I'll try to hit the most important points there.
Traction
Olympus 4.0 equips the most recent, so-called "Vibra Mega" grip technology of Altra. Compared to the outdated "DuraTrac" of Olympus 3.5's, this is a HUGE step forward. Going from 3.5 to 4.0, the lugs also seem to have increased especially through the mid-foot.
"MaxTrac" outsole of Lone Peak 4.5's is undoubtedly better than Olympus 3.5's "DuraTrac".

However, "Vibra Mega" of Olympus 4.0 still shines thorough even when it's compared to the "MaxTrac" of Lone Peak 4.5's - owing to its extra capability that somehow "sticks" to the ground.
But... that's the case when things are kept technically.
In practice, with everything else being equal, higher stack height usually also means lesser traction in many cases. Extra cushion that the Olympus 4.0 has over the Lone Peak 4.5's evens out its advantage, resulting in more or less equal traction capabilities in practice.
So, in traction department, it's safe to say Olympus 4.0 = Lone Peak 4.5 > Olympus 3.5 for most of the cases.
Conclusion
I'd highly recommend you to base your decision primarily off the cushion differences.
If you're going to be participating to an ultra-marathon, absolutely check out the event distance & the chart in the introduction and pick accordingly.
In summary, with everything else being equal and average, if the event distance is more than, say, 75 miles (120 km); then the Olympus is likely to be a better choice over the Lone Peak's because of its extra cushion.
If you won't be participating to ultramarathons...
In summary, with everything else being average; if you're mostly going to be backpacking, trail running, walking and/or standing uninterruptedly, for long periods of time; then you're highly likely to benefit a lot from the extra cushion Olympus has over the Lone Peak's.
In Altra's trail running line, Lone Peak's are the closest you get to a typical trail running shoe and hence a safer choice than the Olympus and is probably a better option for beginner folks.
Men's Lone Peak 4.5 & Olympus 3.5:
No products found.
Women's Lone Peak 4.5 & Olympus 3.5: