Not sure if this is true where you are, but in the US the difference is can be viewed as applied science versus pure science.

Comp sci leans toward the theoretical aspects of computers and computer software. Electrical and electronics engineering, on the other hand, are almost entirely practical, involved in the design and manufacture of hardware and software.

You have to decide which one you are more comfortable with and what your future goals are. Multiple degrees are good if you are going to teach or work in administration/management. If you are looking to do hands-on work, then you need to focus on a particular discipline. Experience becomes much more important than advanced or multiple degrees in that case, other than maybe a masters specific to a particular field.

I didn't go for a double major, rather chose to simply get a degree in Comp Sci. In the end I found I learned almost nothing, and the few challenging courses were all Math courses in disguise. I wish I had just taken Math. Now you're doing a double major, so you have the math end covered, but the Comp Sci part will (for the most part) be introductory level coding with some math sprinkled throughout. For example the courses I found challenging covered set/graph theory and algorithmic complexity analysis/proofs, which was all just lifted from the Math equivalent. Where-as the two 'hard' (aka remotely challenging) Compi-Sci courses I can remember (of all I took) was the compilers course (with the Dragon book ;), and a course (I forget the name) that had us programming in about every language under the sun (from Miranda to Prolog and everything in between). And that was it. The rest was literally a wast of time and money.

That said I don't know if engineering is any better; and I have a rather low opinion of schooling in general...

From everyone I know who did a CS degree, they said it does 3 things for you:

1. It teaches you how to code.
2. It teaches you advanced CS theory.
3. The projects you do during the degree gives you a portfolio you can use to get a job later.

If you can have all that without the degree, why spend the time to get it? There are a lot of real software professionals that got what they needed from that list outside of a university. I'm a mechanical engineer by degree, but I spend my professional time coding. I took a few CS classes, but I learned the rest on my own and built my own portfolio of projects. However, my focus is solving engineering problems with code and not necessarily CS theory, so I chose to do an ME degree instead of CS. For you, it all comes down to what you want to do. Having 2 degrees won't give you any more "respect" than having one if you're applying for a job that doesn't care about your other degree. The only thing 2 degrees gives you other than another piece of paper is more knowledge. But if all you want is the knowledge, why pay the university to give it to you rather than studying it yourself? Unless you need that expensive piece of paper for something, don't pay for it.

Hope that helps.