Is there any difference between $\dfrac{\partial}{\partial x}\left(x^2+y(x)^2\right)$ and $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x^2+y^2\right)$?
$$\dfrac{\partial}{\partial x}\left(x^2+y(x)^2\right) = 2x+2y(x)\dfrac{\partial y(x)}{\partial{x}}$$
$$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(x^2+y^2\right) = 2x+2y\dfrac{\mathrm{d}y}{\mathrm{d}x}$$
Given that $y$ is a function of $x$, they both give a similar-looking answer, but are there any differences between the two?
$\dfrac{\partial}{\partial x}\left(x^2+y^2\right) = 2x\quad$ even if $y$ is function of other variables. For example, if $y=y(t,x,v,...)$ this changes nothing. The partial differentiation with respect to $x$ doesn't take account of what are the variables in $y$, which is the same as saying that it supposes $y=$constant, even if it isn't the case.
For the partial differentiation relatively to a variable , the other variables become non longer variables, but behave as parameters.
$\dfrac{d}{d x}\left(x^2+y^2\right) =\dfrac{\partial}{\partial x}\left(x^2\right)+\dfrac{\partial}{\partial x}\left(y^2\right) = 2x+2y\dfrac{\partial y}{\partial x}\quad$ : the total derivative takes account of that $y$ is non longer constant, but also function of $x$.
NOTE
In order to answer to the comment of Frank Vel :
This is a comment, but too long to be put in the comments section.
You are confused because you don't define without ambiguity the function of SEVERL VARIABLES that you want differentiate partially. So, you confuse $\frac{\partial}{\partial x}$ with $\frac{d}{d x}$
For example, writing $\frac{\partial}{\partial x}(x^2+x)$ is a non sens if you want differentiate partially the function $f(x,y)=x+y$ with respect to $x$, even if $y=x^2$ or anything else. $$\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}(x+y)=1$$ And $\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}(x+y)=1$
On the other hand, if you don't want a partial derivative, but a total derivative this is different : $$\frac{d}{dx}f(x,y)=\frac{\partial}{\partial x}f(x,y)+\frac{\partial}{\partial y}f(x,y)\frac{dy}{dx}$$ $$\frac{d}{dx}(x+y)=\left(\frac{\partial}{\partial x}(x+y)\right)+\left(\frac{\partial}{\partial y}(x+y)\right)\frac{dy}{dx}=(1)+(1)\frac{dy}{dx}=(1)+(1)(2x)=1+2x$$ which is the same as $\frac{d}{dx}(x+x^2)=1+2x$
So, first you have to decide if you want a partial or a total derivative with respect to a variable.
Second, if you want a partial derivative, you have to clearly define a function with several variables. For example, do not write $f(x,y)=(x+x^2)$ which is a function with only one variable. Write $f(x,y)=(x+y)$ and $y=x^2$. This looks the same, but it is very different considering partial derivatives.
Of course, when one is familiar with partial derivation, it is allowed to simplify the writings and avoid all intermediate steps.