The difference is the emphasis: mathematicians focus on proofs, and therefore have to write proofs of theorems that ensure their validity in all cases. Many of these cases are what physicists would call "pathological", and so they basically ignore them when introducing calculus as a tool for students of physics. If you read the table of contents of virtually any book that professes to introduce the "mathematical methods of physics", you will see how different the emphasis is. Since the physicist doesn't care about proofs that involve concepts beyond basic plane geometry and high school algebra, s/he spends a bunch of time introducing a large number of techniques that use no more than algebraic manipulation and limits. You'll find a laundry list of topics, including: tricks of differentiation and integration, the basics of vectors and multivariate calculus (more or less in order to introduce Maxwell's equations), specific examples of Taylor expansions and infinite series (in order to provide asymptotic approximations of functions--usually solutions to differential equations which represent the physics problem you are seeking to solve--which otherwise are unwieldy or impossible to carry out algebraically), some concepts and formulas from "complex variables" (which is a very deep and beautiful subject when studied more systematically, but is also so useful to physicists because of its connection to infinite series, and hence to differential equations), and "special" functions (which may be thought of as important classes of series solutions to differential equation), and most importantly, a little bit of linear algebra theory, which more or less places the majority of these computational tools in a unified framework. (Later on, physicists make use of the an analog of linear algebra, attempting to carry out similar computations in "infinite dimensional" spaces (Hilbert spaces instead of vector spaces), in a subject that mathematicians call functional analysis. Understanding the computational parts of functional analysis is essential for physicists who desire to take their knowledge of problem solving from the domains of mechanics and E&M, and apply them to problems in quantum mechanics, which is founded on functional analysis, and is in large part responsible for the extent to which the subject itself was developed).

The reason the physicist is able to succeed in so completely developing these tools (without even proving any of them) is because s/he is constantly testing their efficacy by trying them out of physics problems. The life of a physics student is more or less an endless game of trying difficult problems, and then invariably ending up resorting to finding approximations in almost all cases.

You might say, then, that where the mathematician spends his or her time looking for watertight proofs of pure concepts about space, logic, and number, from first principles, the physicist spends an equal amount of time thinking about how to approximate differential equations. It's not too surprising that, while in theory there is a great deal of overlap, in practice, the styles of thinking are very different. Part of this is because mathematics is such a vast subject, with more branches than one can care to count, whereas the number of branches used by physicists can more or less be counted on one hand.

In order to go back to a time when mathematics and physics had not yet diverged, you probably have to return to the days of Euler, or even Newton. If you read William Dunham's excellent book, "The Calculus Gallery", in the very first chapter, you will find a discussion of Newton's very basic work on binomial series--something that physicists use almost every day. However, as that book progresses (it goes in chronological order), once you get to Cauchy, who worked in the early 19th century, you begin to see mathematicians turn their focus to questions that no longer address computational questions that could possibly find direct use by physicists, but instead are more or less exist only in the minds of mathematicians.

There certainly is a branch of real analysis, called "classical analysis", which tends to focus more on concrete examples of infinite series, which of course have roots in basic calculus and physics. You can in fact prove a great deal of interesting things about specific infinite series, but you will not invent algebraic geometry or abstract algebra, or be able to fully appreciate the scope of modern 20th century mathematics, if you confine yourself to studying the properties of just one concrete object. If you do, though, you will eventually find yourself studying complex analysis. Somebody who takes this route--that is, to reject the abstract flavor of 20th mathematics--can learn a great deal about mathematics that also happens to be very useful to physicists. On the other hand, you would be missing out on a great deal of fascinating connections between the abstract mathematics of the 20th century, and applications to problems in computer science.

On the other hand, if you really want to study pure mathematics, using proofs, you'll have to be somewhat patient before you can see applications to physics. These applications will come sporatically. Right away, linear algebra is an example of a very basic pure mathematics subject which is absolutely essential to physics. At about the same level is calculus and infinite series. Then, in mechanics, you will probably encounter what physicists call "analytical mechanics", which is an application of a pure mathematics subject called the "calculus of variations". Your understanding of this subject doesn't have to be very deep to start using it in physics, though. At a slightly higher level is complex analysis, which vastly improves your understanding of infinite series. The next application I can think of is quite a bit more advanced; it concerns general relativity, and could be thought of as an advanced setting for multivariable calculus, but which is inspired by the beauty of Euclidian geometry. I am talking about what used to be called "advanced calculus", but now has many different names. The key object of study is what is called a "manifold". A mathematician would call the subject "differential geometry", whereas a physicist would emphasize the use of objects called tensors. At this point, you will begin to see non-trivial mathematics being used in physics, but with the annoyance that physicists continue to rely on computations rather than complete proofs.

Interestingly enough, Michael Spivak, the author of the classic pure mathematics book "Calculus" (mentioned in this thread, by myself and others), is in fact somebody with an interest in mechanics, and happens to be one of the key expositors of differential geometry (see his 5-volume treatise on the subject). To this effect, he has also written an introductory book on mechanics (and is in the process of writing a sequel on the subject of E&M), but with an emphasis and style unlike any mechanics book written for physicists. The book is called "Physics for Mathematicians: Mechanics". It looks deceptively simple in terms of the amount of formula used; however, it will really only be appreciated by mathematicians who have studied some amount of differential geometry (I believe he says in the preface that it would be idea for the read to have read some subset of his 5-volume treatise on the subject). Back in the `60s, Spivak also wrote what, for years, seems to have been definitive text on multivariable analysis and differential geometry for pure mathematicians, in a book called "Calculus on Manifolds". This book is quite difficult to read, though, and today, there are more friendly introductions that presuppose less mathematical maturity, and are less terse (although the book is still a classic).