Newton vs Leibniz notation I have often come across the cursory remarks made here and there in calculus lectures , math documentaries or in calculus textbooks that Leibniz's notation for calculus is better off than that of Newton's and is thus more widely used.
Though I have always followed Leibniz's notation( matter of familiarity, as that's what I have been taught) , but of late I had the idea of following Newton's notation just to see where I could get stuck just because of "notational" issues. 
Is there any limitation of Newton's notation that I might encounter while doing calculus ; and which may make it seem a bad idea to do calculus in Newton's notation?
Here "Leibniz notation" is $\frac{dy}{dx}$  for the derivative of $y$, and "Newton's notation" is $\dot{y}$ for the derivative of $y$.
 A: Gottfried Liebniz developed his calculus around 1673 and published it at 1684, fifty years before Newton's work1 on that subject was posthumously published. That could be one of the reasons why it is more widely used.
Additionally, that time difference in writing and publishing became a subject of rivalry over who of the two mathematicians first developed calculus, one of the direct implications of that conflict was the use of notation from the respective followers, leading to many difficulties for further developing of calculus in England2 for many years.
Nowadays, both notations are being used interchangeably depending on the stage of the solution of the equation involving derivatives, for example: for algebraic manipulations one can use the more brief Newton notation, but when the time comes to separate the variables one writes the terms using Liebniz notation. Newton's notations (for derivatives) specifically is being more widely used in, mechanics, electrical circuit analysis and more generally in equations where differentiation is more obvious.

1. Method of Fluxions is the book in which Newton describes differential calculus and it was completed in 1671, but published in 1736.
2. An opinion described in Men of Mathematics by E. T. Bell
A: Wikipedia has a dedicated page on notations for differentiation, in short:


*

*Leibniz $\frac{dx}{dt}$

*Newton $\dot{x}$

*Lagrange $x'(t)$
Leibniz's notation is suggestive, thanks to the cancelling of the differentials
in the chain rule:
$$
\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}
$$
however great care must be taken, as this notation can also be misleading for higher order derivatives:
$$
\frac{d^2y}{dt^2}=\frac{d^2y}{dx^2}\frac{dx^2}{dt^2}=\frac{d^2y}{dx^2}\left(\frac{dx}{dt}\right)^2
$$
which is wrong, the right formula is:
$$
\frac{d^2y}{dt^2}=\frac{d^2y}{dx^2}\left(\frac{dx}{dt}\right)^2+\frac{dy}{dx}\frac{d^2x}{dt^2}
$$
You have not this problem with Lagrange's notation:
$$
y(x(t))''=(y'(x(t))x'(t))'=y''(x(t))(x'(t))^2+y'(x(t))x''(t)
$$
These notation problems are well known when teaching differential calculus, see:
H. Poincaré, La Notation Différentielle et l'enseignement (pdf)
J. Hadamard, La notion de différentielle dans l'enseignement (pdf)
unfortunately both in French, however you can find an English translation of Hadamard's article here.
You can also see:
Differentials, higher-order differentials and the derivative in the Leibnizian calculus (pdf)
A: Regarding the notations for the derivative:
Upsides of using Leibniz notation: 


*

*It makes most consequences of the chain rule "intuitive".  In particular, it is easier to see that $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ than it is to see that $[f(g(x))]' = f'(g(x))\cdot g'(x)$.  See also $u$-substitution, in which we "define $du := \frac{du}{dx}dx$".

*In a physical/scientific setting, it makes it obvious what the units of the new expression (integral or derivative) should be.  For instance, if $s$ is in meters and $t$ is in seconds, clearly $\frac{ds}{dt}$ should be in meters/second.


Downsides:


*

*It is harder/clumsier to keep track of arguments of the derivative with this notation.  For instance, I can more easily write and keep track of $f'(2)$ than I can $\left.\frac{dy}{dx} \right|_{x=2}$

*It often leads to the mistaken notion that $\frac{dy}{dx}$ is a ratio
Notably, almost no one uses Newton's notation for the integral ("antiderivative"), in which the antiderivative of $x(t)$ is $\bar x(t)$, $\overset{|}{x}(t)$, or $X(t)$ (though this last one occasionally is used in introductory textbooks).  Leibniz notation seems to be the clear winner in that regard.
A: The most obvious difference is that the Leibnitz notation strictly defines what the independent variable is. In basic calculus we tend, as a rule, to derive a function "y" of a variable "x", but what happens when you want to derive the function
$w = 3x+4m$?
How would the Newton notation help you understand which is the variable and which is the parameter?
Also, in integrals, the notation makes methods like substitution or integration by parts much simpler as you use the "dx" symbol as if it were a substitutable variable.
A: I think it's best to use both notations simultaneously.
For instance, my preferred statement of the chain rule is:
$$\frac{d}{dx}f(y) = f'(y)\frac{d}{dx} y$$
For example, we can write:
$$\frac{d}{dx} \sin(x^3) = \sin'(x^3)\frac{d}{dx}x^3 = \cos(x^3)\cdot 3x^2 = 3x^2 \cos(x^3)$$
Try to do this using just Newtonian notation, or just Leibnizian notation; you'll quickly notice that both are harder.
There's also multivariable versions. For instance:
$$\frac{d}{dt}f(x,y) = (D_0 f)(x,y) \frac{d}{dt} x+(D_1 f)(x,y) \frac{d}{dt} y$$
