What is the difference between a point and 2 points that are separated by an infinitesimal distance? Let's say that over here we have 1 point on some coordinate axis, by itself.
And way over here, we have 2 points that were originally separated by a distance of 1, but we shrunk that distance down to an infinitesimal length. 
What then is the difference between these 2 scenarios? Can the 2nd case be treated as 1 point?
 A: The OP seems to be struggling with the notion of limit, when for example a point $A=1$ on the real axis is being approached by a variable point $B=1+\frac{1}{n}$ so that for $n=1$ the point $B$ is located at distance $1$ from $A$ and as $n$ increases, $B$ gets closer and closer to $A$.  To speak loosely, for infinite $n$ it seems that $B$ is infinitely close to $A$ or more precisely the distance between $A$ and $B$ is infinitesimal, but still not zero. Where does the limit come in, and how can you claim that in the limit $A$ and $B$ are the same?
The above is my understanding of the OP's query. The answer is provided conveniently in the hyperreal framework by the standard part function.  The limit is not the value of $B$ for an infinite index $n$ but rather the standard part thereof.  Only after taking the standard part can one claim that the points are the same.  This is a more direct interpretation of the limit concept than in the epsilon, delta approach where the value of the limit needs to be given in advance, rather than computed directly.
For an accessible introduction see Keisler's Elementary Calculus.
A: It appears to me that the answer is in your question. As you are asking about points with an infinitesimal distance between them, you are working within a number system that includes infinitesimals. See Wikipedia for examples of such number systems. Within such a number system, the points are distinct and are separated by an infinitesimal distance. Their "real number part" however, is the same (much as the "whole number part" of 3.2 and of 3.6 is the same, although as decimals they are different numbers).
A: The elements of sets $X$ which are considered to be "spaces" are called points. A point is a point, and is different from all other points in the same space. If a distance function (or: metric) $d$ is defined on $X$ then $d(p,q)$ is a positive real number whenever $p\ne q$. There is no such thing as "infinitesimally close" points in this context.
On the other hand, if such a point $p$ arises as part of a larger configuration under study you may want to qualify the "standing" of this point within this configuration. An example: The curves $y=x$, $y=x^2$, and $y=x^3$ all intersect the $x$-axis at the single point  $p:=(0,0)$, but the quality of this intersection is different in the three cases. As a mathematician you want to come to grips with this distinction. You then say that $y=x$ has a simple zero at $x=0$, $y=x^2$ a double zero, and so on. After some analysis you will propose the definition that a function $f$ has an $n$-fold zero at $x_0$if there is an $a\ne0$ such that
$$f(x)-f(x_0)=(x-x_0)^n\bigl(a+o(1)\bigr)\qquad(x\to x_0)\ .$$
In colloquial language one talks about an $n$-fold tangency between the graph of $f$ and the $x$-axis in such a case. Sometimes one is even inclined to call an (ordinary) tangent point a "double"  point, e.g., in connection with counting the solutions of quadratics.
