Difference between gradient and Jacobian

These are two particular forms of matrix representation of the derivative of a differentiable function $f,$ used in two cases:

• when $f:\mathbb{R}^n\to\mathbb{R},$ then for $x$ in $\mathbb{R}^n$, $$\mathrm{grad}_x(f):=\left[\frac{\partial f}{\partial x_1}\frac{\partial f}{\partial x_2}\dots\frac{\partial f}{\partial x_n}\right]\!\bigg\rvert_x$$ is the matrix $1\times n$ of the linear map $Df(x)$ exprimed from the canonical base of $\mathbb{R}^n$ to the canonical base of $\mathbb{R}$ (=(1)…). Because in this case this matrix would have only one row, you can think about it as the vector
  $$\nabla f(x):=\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\dots,\frac{\partial f}{\partial x_n}\right)\!\bigg\rvert_x\in\mathbb{R}^n.$$
  This vector $\nabla f(x)$ is the unique vector of $\mathbb{R}^n$ such that $Df(x)(y)=\langle\nabla f(x),y\rangle$ for all $y\in\mathbb{R}^n$ (see Riesz representation theorem), where $\langle\cdot,\cdot\rangle$ is the usual scalar product
  $$\langle(x_1,\dots,x_n),(y_1,\dots,y_n)\rangle=x_1y_1+\dots+x_ny_n.$$
• when $f:\mathbb{R}^n\to\mathbb{R}^m,$ then for $x$ in $\mathbb{R}^n$, $$\mathrm{Jac}_x(f)=\left.\begin{bmatrix}\frac{\partial f_1}{\partial x_1}&\frac{\partial f_1}{\partial x_2}&\dots&\frac{\partial f_1}{\partial x_n}\\\frac{\partial f_2}{\partial x_1}&\frac{\partial f_2}{\partial x_2}&\dots&\frac{\partial f_2}{\partial x_n}\\ \vdots&\vdots&&\vdots\\\frac{\partial f_m}{\partial x_1}&\frac{\partial f_m}{\partial x_2}&\dots&\frac{\partial f_m}{\partial x_n}\\\end{bmatrix}\right|_x$$ is the matrix $m\times n$ of the linear map $Df(x)$ exprimed from the canonical base of $\mathbb{R}^n$ to the canonical base of $\mathbb{R}^m.$

For example, with $f:\mathbb{R}^2\to\mathbb{R}$ such as $f(x,y)=x^2+y$ you get $\mathrm{grad}_{(x,y)}(f)=[2x \,\,\,1]$ (or $\nabla f(x,y)=(2x,1)$) and for $f:\mathbb{R}^2\to\mathbb{R}^2$ such as $f(x,y)=(x^2+y,y^3)$ you get $\mathrm{Jac}_{(x,y)}(f)=\begin{bmatrix}2x&1\\0&3y^2\end{bmatrix}.$