As an engineer who is greatly interested in maths methods in physics, I will leave some comments.
Both are anyway intended to provide students with sufficient mathematical background for learning their majors. So the greatest difference should come from the applications. Different emphases on applications make huge differences in the writing styles.
I guess the core topics for physics undergrads start with electrodynamics and go onto quantum mechanics. (Let's not talk about classical mechanics and statistical mechanics for now.) So the goal of maths methods classes should be to understand vector calculus and linear algebra. To be more specific, the ultimate goal is to be able to apply Gauss's and Stokes' theorem, and to be able to solve eigenvalue equations. Also, a lot of special functions are there to be used in solving physics problems.
I will have to make clear that what's typically called engineering maths, in comparison to maths methods for physics, is about the maths used in pure engineering subjects - which hardly depend on knowledge of physics. It's basically (a) what electrical and information engineers do, (b) what industrial engineers do, and (c) what computer and data scientists do.
(a) includes circuit analysis, digital electronics, signal processing, control theory, etc. Such subjects apply differential equations, Laplace transforms, Z transform and Fourier transform. However, you know, DE theory has really extensive applications, hence is used throughout engineering. So this is a core topic of a typical engineering maths book.
(b) includes linear programming, production process analysis, financial engineering, etc. All of such subjects direct towards the optimisation methods. Note that both (a) and (b) use a lot of specialised diagrams - unrelated to physics - to model various real-world phenomena and processes.
(c) includes algorithms, data structure, some other subjects common with information engineering, software production theory, machine learning, etc. In doing these, the knowledge of discrete mathematics is essential. The latter requires linear algebra, multivariable calculus and statistics.
So, a good engineering maths book, at least as a reference, should include most of the topics mentioned above. But note that, unlike physics majors in which some hard maths methods books really provide all maths you will ever encounter in an undergrad course, an engineering maths book usually touches the minimum.
To talk about the philosophical aspects, this is because physics has an ultimate goal of explaining the world in unified and concise mathematical laws, whereas engineering is much broader and thus it's impossible to set a single goal. So the maths used depends on a sub-field of study, and specific mathematical techniques are usually taught within engineering courses themselves.
To be honest, there are two aspects of engineering - an applied maths aspect and an applied physics aspect. All of the things I mentioned so far are the applied maths aspect. It looks fair to teach applied maths only in a separate applied maths class, not in a single engineering maths class. For example, ocean engineers needn't really learn about the maths of signal processing. It's like maths methods for physics courses not really going deep into the details of variational calculus and perturbation theory, tensor analysis and differential geometry, group and representation theory, etc.
But the applied and experimental physics aspect is also a huge part of engineering. Actually, it has a closer meaning to what engineering is classically and originally supposed to be - like civil engineering which has originated from constructing scientific ways of building housings and facilities, or like chemical engineering that was essential for large-scale production of various useful substances and drugs.
Such subjects require a deep enough understanding of physical principles. Electrical engineers must know how electrodynamics works, mechanical engineers must know how fluid dynamics works, chemical engineers must know how thermodynamics works, materials engineers must know how quantum mechanics works. Obviously, they have to learn the maths needed for individual physics subjects. In that case, maths methods for physics books will be really good references for engineers.
I think a good engineer is someone who has a solid and fundamental understanding of at least one and ideally two or more sub-fields of engineering. That means he knows the basic mathematical and physical principles and understands how engineering works in industries. Then whatever new problem comes to the front of him, he may quickly learn, adapt and eventually resolve the problem.
Finally, the reason why I am talking about all these is that I don't think it's not a simple matter of distinguishing between books. Like all other things, engineering is a very broad and complexly interrelated subject, so it's quite meaningless to talk about the curriculum of the discipline as a whole.
But I can guarantee that a typical engineering maths book should prepare you for the bare minimum of mathematical techniques. Actually, if it's the first time that you are learning university maths - especially ODE - then I would strongly recommend using an engineering maths book, whether you are an engineer, a physicist, or a chemist.
