When should I use "graph" vs. "plot"? My ODEs textbook uses both graph and plot but I can't figure out how it chooses one over the other.
From the book:

Sketch the graph of the solution in the x1x2-plane for t ≥ 0.
  [this one was referring to a continuous function]

Also from the book:

Plots of the solution and a tangent line approximation for the initial value problem (11).
  [this one referring to a continuous solution]

Is there a formal, to some degree, distinction between the two terms? When do I call it a plot and when do I call it a graph?
 A: 
we plot points, but we graph functions

this is useful guidance since:
"plot"  is to draw a graphic representation with respect to measurements or coordinates
"graph" is to illustrate connections between several things by plotting dots and lines
A: A graph in the sense of the object "a graph of a function" often has the specific definition, for a real-valued function $f$ of being the set of points $\{(x,y)|y=f(x)\}$.  That is, graph can be used as a noun to mean literally the set of ordered pairs.  It is interesting to note that this is also sometimes the definition of the function itself.
Beyond that one specific technical meaning, I do not think that (in the context of ODEs, calculus, or precalculus mathematics) there is a formal distinction between the two words.  Informally, I use each word a bit differently.
The word graph in the sense of the action "graph a function" means to make a drawing of the set of points in the (noun) graph of the function.  This sometimes has the connotation of a careful, precise drawing.  In contrast, "sketch the graph" sometimes has the connotation of being a less-formal illustration showing key features of the graph of the function without necessarily being as precise.
The word plot in the sense of the object "a plot of a function" means a visual representation.  There are other types of plots, such as scatter plots and line plots, that would not typically be referred to as graphs.  I would say that graphs are a specific type of plot, but I'm not sure that's quite right.
The word plot in the sense of the action "plot the graph of a function" or "plot some points" typically means drawing with the connotation of precision.  I generally would not use the phrasing "plot the graph."
A: *

*'Plot' (verb) has a more active, tangible connotation than 'graph' (verb), ie a person or machine placing ink on paper, a surveyor doing measurements and markings, a computer monitor radiating visible light producing some representation or figure, , and so on.  Obviously, 'plot' is associated with 'plotting devices' which evokes all sorts of technologies.  I would not be surprised if the word 'plot' has its roots in surveying, navigation, and astronomy, in that order.  

*'Graph' (verb) has a more abstract intangible connotation than 'plot'.  When this word is used the resulting 'graph' (as a noun) is by design a representation of some mathematical object (see below).  And this object is the goal, not the representation itself.  For example, a teacher may graph $y = x^2$ on a dusty blackboard, but once he starts talking about the graph, it's usually not the chalk he's talking about anymore, he's talking about a mathematical object.

*'Graph' (noun) is closely related to 'function'.  In mathematics, functions are often equated with their graphs, IE a function $f: X \rightarrow Y$ is just the choice of two sets $X$ and $Y$ and an appropriate subset of $f \subset X \times Y$, the latter being 'the graph' in the formal sense.  When visualized in the usual way, with elements from X as the independent variables, we can call these the traditional graphs.

*Finally, from wikipedia: graphs, plots, and charts.  In the first link,  graphs are of the 'traditional' sort.  In the second link, a large number of plot types are given, including box plots and scatter plots.  However, in addition there are many graphs of the traditional sort, but with modified/scaled axes.  The entry on charts also has many examples which may be called either graphs or plots.
A: As per @Alain Pannetier's advice, I'm making my comment into an answer. (To lazy to rephrase now, but I'll most likely try to make it pretty and more answer-y later.)
...In all of my math classes (I'm a math major), we talk about plotting points, but we graph functions. You can plot specific points in the graph of a function, but you don't just plot a function. Generally, plot, as a noun, refers to a set of points that may or may not be connected by a line, but that cannot be represented as a function. I don't know if there's some technical ground behind this, but if there is, one of us is being confused...
So, in short, "plot" is used for a finite set of points, while a "graph" is used for a function comprised of infinite points.
A: Although I am not a native English speaker, I believe plotting refers to drawing a diagram of any kind. So you can plot a function, an architectural design , a software class diagram etc.
Graph is a more specific term that refers to plotting by using a system of axes. So while both are correct, graph is more precise. 
To add to the confusion, there are also the notion of graph from graph theory. In my native language (Greek) we use graph only in relation with graph theory and grapheme for the drawing/plot/graph of a function. However, grapheme has a different meaning in English.
In conclusion, you could call a quick drawing on a blackboard that is not very precise a plot but not a graph. A precise drawing using axes can be called a graph and a plot.
