All that's going on here is that when you differentiate a Taylor series expansion
$$f(x) = f^{(0)}(a) + \frac{f^{(1)}(a)}{1!} (x - a)^1 + \frac{f^{(2)}(a)}{2!} (x - a)^2 + ...$$
you get the Taylor series expansion
$$f'(x) = f^{(1)}(a) + \frac{f^{(2)}(a)}{1!} (x - a)^1 + \frac{f^{(3)}(a)}{2!} (x - a)^2 + ....$$
In other words, what you've done is precisely shifted the Taylor coefficients one to the left. That's all the Wikipedia article is saying.
I disagree that this makes ODEs any easier to solve, though. ODEs were already this easy to solve, you just weren't taught the right language for seeing this.
The flow is an important idea in the sense that it allows us to examine the behavior of solutions to an ODE at a "larger scale." What I mean by this is that instead of tracking a single solution, emanating from a point, we can track an ensemble of solutions, emanating from a set of points, all at the same time, and we can use this to study how the ODE distorts these points.
This is particularly important in applications of ODEs in PDE, and especially important in continuum mechanics. For instance, let's consider the following transport equation:
$$
\partial_t u(t,x) + a(x) \cdot \nabla u(t,x) =0
$$
where $a: \mathbb{R}^n \to \mathbb{R}^n$ is Lipschitz. If we let $\varphi : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ denote the flow map associated to $a$ then we have that
$$
\frac{d}{dt} u(t,\varphi(t,x)) = \partial_t u(t,\varphi(t,x)) + \partial_t \varphi(x,t) \cdot \nabla u(t,\varphi(t,x)) \\
= \partial_t u(t,\varphi(t,x)) + a(\varphi(t,x)) \cdot \nabla u(t,\varphi(t,x)) =0
$$
which tells us that $u$ is "constant along the flow." In particular, if we specify the initial condition $u(0,x) = g(x)$, then
$$
u(t,\varphi(t,x)) = u(0,\varphi(0,x)) = u(0,x) = g(x)
$$
and hence we can solve for $u$ via
$$
u(t,x) = g(\varphi^{-1}(t,x)) = g(\varphi(-t,x)).
$$
This establishes a key relationship between the flow of a vector field and the "transport" operator / "convective derivative" induced by the vector field $a$. This plays an essential role in continuum mechanics in going back and forth between the material (Lagrangian) and laboratory (Eulerian) coordinate frames.
Since you already know that the flow map induces a $1-$parameter family of diffeomorphisms, I would suggest studying flows by trying to prove another one of the main theorems about them, Liouville's theorem. It says that
$$
\det D \varphi(t,x) = \exp\left( \int_0^t \text{div}f(\varphi(s,x)) ds \right)
$$
whenever $\varphi$ is the flow associated to $f$. This is another extremely important result in continuum mechanics, as it builds a link between incompressibility (the divergence free condition) and the local preservation of volume. Indeed, if $\text{div} f=0$, then Liouville shows that
$$
\det D \varphi(t,x) =1 \text{ for all }t,x,
$$
and so if $U \subseteq \mathbb{R}^n$ is a measurable set, then
$$
|\varphi(t,U)| = \int_{U} |\det D \varphi(t,x)| dx = \int_U dx = |U|,
$$
which tells us that the volume, or Lebesgue measure, of $U$ is preserved along the flow for all measurable sets.
Best Answer
I think you have done some mistakes when writing down your equations. An implicit ordinary differential equation can be written: $$f(t,x,x') = 0$$ and a DAE can be written: $$f(t,x,x') = 0$$ and since these two equations are syntactically equal, it is very easy to be confused about the distinction.
First, I want to make one thing clear. When talking about ODEs it is quite alright to talk about just one equation, e.g. the equation $$y'(t) + y(t) + t = 0$$ is an ordinary differential equation.
But when talking about a DAE, you always, in a non-trivial case, talk about a system of equations. If your DAE contains only one equation it will either be a differential equation or an algebraic equation (in this context, algebraic means not containing any derivatives). Thus, in the rest of the post, $f$ and $y$ will be vectors of functions.
Thus, the difference between an implicit ODE system and a DAE system is, in a way, that the DAE system can contain purely algebraic equations and variables. The more technical and correct criterion is that the Jacobian $$\frac{\partial f(t,x,x')}{\partial x'}$$ needs to be non-singular for the system $f(t,x,x') = 0$ to be classified as an implicit ODE.
To make the distinction more clear between a DAE and an implicit ODE, you can split the vector $x$ in two parts, $x_D$, containing the $x$ for which derivatives occur in the DAE and $x_A$, containing the algebraic $x$, i.e. the $x$ for which no derivative occur in the DAE, and we write the DAE on the form $$f(t,x_A,x_D,x_D') = 0.$$ we can also split the function $f$ into two parts: $f_D$ containing the equations containing derivatives and $f_A$ not containing any derivatives.
A classic example of a DAE is the following formulation for the motion of a pendulum: $$\begin{align} 0 &= x' - u \\ 0 &= y' - v \\ 0 &= u' - \lambda x \\ 0 &= v' - \lambda y - g\\ 0 &= x^2 + y^2 - L^2 \end{align}$$ where $L$ (the length of the pendlum) and $g$ (gravitational acceleration) are constants. Classifying the variables as differential (belonging to $x_D$) and algebraic (belonging to $x_A$), we see that $x,y,u,v$ are differential and $\lambda$ is algebraic. All equations except the last (the length constraint) are differential.
We can calculate the Jacobian of this system. We order the functions as above and the variables as follows: $(x,y,u,v,\lambda)$. Then the Jacobian will be: $$\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$$ which is singular since the last row ad the last column is zero, hence the system is a DAE.
One often wants to apply techniques to a DAE to transform it to a semi-explicit DAE of index 1, which can be written as follows: $$\begin{align} x_D' &= g_D(t,x_D,x_A) \\ 0 &= g_A(t,x_D,x_A) \end{align}$$ because then $g_A$ can, in theory, be solved for $x_A$, which can then be inserted into $g_D$, which can then be integrated numerically.
The pendulum example might look as it is on this form, but its index is not 1.