There's a difference in nuance. Literally, a "landmark" is something on land that you can see from a ship and use for navigation. Thus, it would be a tall structure, easily seen from far away, and easily recognized. By extension, it can mean something that people who live in an area can see from far away and use for orientation. For example, in Boston, the Bunker Hill monument (which looks like an obelisk) is a landmark, and the John Hancock Tower, the tallest building in Boston, is another. Literally, a "milestone" is a stone marker along a road that tells you how many miles you are from someplace. Thus, passing milestones is a quantitative measure of progress. Figuratively, "milestones"--for example, in a business or engineering project--are identifiable events that can be used to as a measure of progress. They might not be particularly big or impressive, and there would be a quite a number of them--as regularly spaced as possible--so that the manager can gauge how the project is going, and estimate how long it will take. "So far, we've reached all of our milestones on time, so I think the project will be completed on time." A landmark event means one that is big, visible, and, if you like, can be seen "from a distance." One might say, "In the Apollo space program, every flight was a milestone, but in everybody's minds the landmark event was Apollo 11, the first moon landing."

"Milestone" can refer to a physical object, but I have only ever heard it used in reference to an event. Examples of events typically referred to as "milestones" are things like graduation, wedding, birth of your children, etc. Here are some sentences in which "milestone" is used to refer to an event: "Her college graduation was an important milestone in her life because it meant that she was done with school for good." "Our wedding was a milestone that we remember every year by celebrating our anniversary." "We wanted to capture every milestone in Susan's childhood, so we started taking photos of her since she was a baby." "Finishing his first marathon was a milestone in his athletic career." Usually, "landmark" refers to a specific object or part of a landscape that helps people remember something about that place. Landmark can also refer to an event, however it is usually used to discuss something historically significant rather than personally significant. (Milestone can also be used to refer to something historically significant.) I've also heard "landmark" used to describe important court cases. Here are some examples in which "landmark" is used to describe or refer to an event: "Brown v. Board of Education of Topeka was a landmark decision that imporved educational opportunities for students all over the country." "Gettysburg was a landmark battle that many consider the turning point of the Civil War." "The first manned landing on Mars would surely become a landmark in aerospace history." (In this case, we could also say "The first manned landing on Mars would surely become a milestone in aerospace history.") "Milestone" is a noun, so we cannot replace "landmark" with "milestone" in the first two sentences.

A landmark can be anything that is obvious in the area - a building, mountain, river, lake, and so on. It's easy to see and you can use it to navigate or orient yourself, because it "marks" the land. A milestone is exactly that: a stone which shows the distance (in miles) from a given point, eg. from a city centre. Both words can be used figuratively.