Are "differential forms" an algebraic approach to multivariable calculus?

Any approach to multivariable calculus necessitates some modicum of multilinear algebra ( for instance because the change of variables formula for integrals uses determinants), and the cleaner the interface between calculus and algebra, the better. The language of differential forms is a clean interface between the two, and this language generalizes from $\mathbb{R}^n$ to arbitrary manifolds. That is, differential forms are how calculus is done on manifolds. Or, to be more accurate, they are a convenient algebraic formalism for doing calculus on manifolds. To abandon differential forms is not to go to analysis, but to lose oneself in an ocean of notation.

The answer to the question

• Are differential forms the only approach to multivariable calculus?

is a definitive NO. Differential forms are a topic of differential geometry or calculus on manifolds. And I think Wikipedia takes the right approach to calculus on manifolds by first talking about implicit and inverse function theorems, vector fields, the directional derivative, the Lie derivative, the Lie bracket and other important topics before even mentioning differential forms.

If you define multivariable calculus as the extension of calculus in one variable to calculus in more than one variable, and divide calculus into differential calculus and integral calculus, an introduction to differential forms as part of multivariable integral calculus could make sense.

• Are "differential forms" basically an algebraic approach to (multivariable) calculus?

It's true that differential forms have important algebraic properties that are useful for the global analysis of manifolds:

• Differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism. (A vector field can only be pulled back by an immersion if it's tangential to the immersed manifold.)
• The exterior derivative has the important property that $d^2 = 0$.

This leads to exact sequences which allow to build algebraic cohomology theories.

• What would be a analysis counterpart and what are the advantages and disadvantages of these two approaches?

One drawback of differential forms for high dimensional manifolds comes from the curse of dimensionality. The vector space of alternating $k$-linear forms over an $n$-dimensional vector space has dimension $\frac{n!}{k!(n-k)!}$. So failing to visualize differential forms "correctly" before your inner eye is not the fault of your imagination, but simply one of the disadvantages of differential forms. As a consequence, there is nothing wrong with visualizing differential forms in $\mathbb{R}^3$ as vector fields.

Perhaps surprisingly, there are ways around the curse of dimensionality. (Monte Carlo methods are probably the best known examples, but there also exist deterministic methods.) However, there is so much material to be covered in a multivariable calculus course that the curse of dimensionality is rarely even mentioned at all.