What is the difference between a point and a vector?

Here's an answer without using symbols.

The difference is precisely that between location and displacement.

• Points are locations in space.
• Vectors are displacements in space.

An analogy with time works well.

• Times, (also called instants or datetimes) are locations in time.
• Durations are displacements in time.

So, in time,

• 4:00 p.m., noon, midnight, 12:20, 23:11, etc. are times
• +3 hours, -2.5 hours, +17 seconds, etc., are durations

Notice how durations can be positive or negative; this gives them "direction" in addition to their pure scalar value. Now the best way to mentally distinguish times and durations is by the operations they support

• Given a time, you can add a duration to get a new time (3:00 + 2 hours = 5:00)
• You can subtract two times to get a duration (7:00 - 1:00 = 6 hours)
• You can add two durations (3 hrs, 20 min + 6 hrs, 50 min = 10 hrs, 10 min)

But you cannot add two times (3:15 a.m. + noon = ???)

Let's carry the analogy over to now talk about space:

• $(3,5)$, $(-2.25,7)$, $(0,-1)$, etc. are points
• $\langle 4,-5 \rangle$ is a vector, meaning 4 units east then 5 south, assuming north is up (sorry residents of southern hemisphere)

Now we have exactly the same analogous operations in space as we did with time:

• You can add a point and a vector: Starting at $(4,5)$ and going $\langle -1,3 \rangle$ takes you to the point $(3,8)$
• You can subtract two points to get the displacement between them: $(10,10) - (3,1) = \langle 7,9 \rangle$, which is the displacement you would take from the second location to get to the first
• You can add two displacements to get a compound displacement: $\langle 1,3 \rangle + \langle -5,8 \rangle = \langle -4,11 \rangle$. That is, going 1 step north and 3 east, THEN going 5 south and 8 east is the same thing and just going 4 south and 11 east.

But you cannot add two points.

In more concrete terms: Moscow + $\langle\text{200 km north, 7000 km west}\rangle$ is another location (point) somewhere on earth. But Moscow + Los Angeles makes no sense.

To summarize, a location is where (or when) you are, and a displacement is how to get from one location to another. Displacements have both magnitude (how far to go) and a direction (which in time, a one-dimensional space, is simply positive or negative). In space, locations are points and displacements are vectors. In time, locations are (points in) time, a.k.a. instants and displacements are durations.

EDIT 1: In response to some of the comments, I should point out that 4:00 p.m. is NOT a displacement, but "+4 hours" and "-7 hours" are. Sure you can get to 4:00 p.m. (an instant) by adding the displacement "+16 hours" to the instant midnight. You can also get to 4:00 p.m. by adding the diplacement "-3 hours" to 7:00 p.m. The source of the confusion between locations and displacements is that people mentally work in coordinate systems relative to some origin (whether $(0,0)$ or "midnight" or similar) and both of these concepts are represented as coordinates. I guess that was the point of the question.

EDIT 2: I added some text to make clear that durations actually have direction; I had written both -2.5 hours and +3 hours earlier, but some might have missed that the negative encapsulated a direction, and felt that a duration is "only a scalar" when in fact the adding of a $+$ or $-$ really does give it direction.

EDIT 3: A summary in table form:

+--------------------+------------------------+-----------------------+
| Concept            | SPACE                  | TIME                  |
+--------------------+------------------------+-----------------------+
| LOCATION           | POINT                  | TIME                  |
| DISPLACEMENT       | VECTOR                 | DURATION              |
+--------------------+------------------------+-----------------------+
| Loc - Loc = Disp   | Pt - Pt = Vec          | Time - Time = Dur     |
|                    | (3,5)-(10,2) = <-7,3>  | 7:30 - 1:15 = 6hr15m  |
+--------------------+------------------------+-----------------------+
| Loc + Disp = Loc   | Pt + Vec = Pt          | Time + Dur = Time     |
|                    | (10,2)+<-7,3> = (3,5)  | 3:15 + 2hr = 5:15     |
+--------------------+------------------------+-----------------------+
| Disp + Disp = Disp | Vec + Vec = Vec        | Dur + Dur = Dur       |
|                    | <8,-5>+<-7,3> = <1,-2> | 3hr + 5hr = 8hr       |
+--------------------+------------------------+-----------------------+

Points and vectors are not the same thing. Given two points in 3D space, we can make a vector from the first point to the second. And, given a vector and a point, we can start at the point and "follow" the vector to get another point.

There is a nice fact, however: the points in 3D space (or $\mathbb{R}^n$, more generally) are in a very nice correspondence with the vectors that start at the point $(0,0,0)$. Essentially, the idea is that we can represent the vector with its ending point, and no information is lost. This is sometimes called putting the vector in "standard position".

For a course like vector calculus, it is important to keep a good distinction between points and vectors. Points correspond to vectors that start at the origin, but we may need vectors that start at other points.

For example, given three points $A$, $B$, and $C$ in 3D space, we may want to find the equation of the plane that spans them, If we just knew the normal vector $\vec n$ of the plane, we could write the equation directly as $\vec n \cdot (x,y,z) = \vec n \cdot A$. So we need to find that normal $\vec n$. To do that, we compute the cross product of the vectors $\vec {AB}$ and $\vec{AC}$. If we computed the cross product of $A$ and $C$ instead (pretending they are vectors in standard position), we could not get the right normal vector.

For example, if $A = (1,0,0)$, $B = (0,1,0)$, and $C = (0,0,1)$, the normal vector of the corresponding plane would not be parallel to any coordinate axis. But if we take any two of $A$, $B$, and $C$ and compute a cross product, we will get a vector parallel to one of the coordinate axes.

In spirit they are different things. But the usual convention is to think of vector in the plane or in three-dimensional space as starting at the origin. In that case, a vector is identified precisely by its ending point, giving you an identification between points and vectors.

One way to see that they are different things (even if identified in many circumstances), is that you can add vectors, while the sum of points makes no sense. Same with the dot and cross products.