The big downside to C++ on iOS is, well, XCode support for C++ stinks. I would go far as to say XCode stinks, but some people seem to like it for reasons I can't quite fathom.

What? XCode has very good C++ "support", using LLVM/Clang for compilation and static analysis. They've a quite nice integration going there, really, so where are you getting your facts? And that last part -- I like XCode, in fact, I prefer it highly over Visual Studio and I'm a professional (if that lends some kind of magical power to my opinion). If you can't fathom that, I'd say you're far from reality and just limiting yourself out of some illogical reason.

/Going way OT

... alright, riddle me this then. Describe the process of creating a C++ game in XCode... what options do you pick? This perfectly illustrates the importance of C++ in XCode.

Most of the information from all answers in this thread is complete bullshit and has nothing to do with the OPs origional request which is what are good tools for developing iOS and Android Apps and not one single post has tried to answer his question.

If you are a complete beginner then using the languages and software recommended by the manufacturers is the way to go. This means that:
For iPhone you should use Xcode and Objectivve C.
For Android you should use Eclipse and Java.

If you are wanting to get your foot wet in doing cross platform games then something like Construct2 or Game Maker or even Unity.

If you are more experienced and a lot more code orientated and want to make games then something like moai or another engine that I can really recommend gameplay3D from rim.

Of course if your apps are not going to be games then really just stck with the manufacturers recommendations. THere are other solutions like html5 engines or flex to iOS but if you get stuck trying to sign an app then you have almost no support. Genuinly if you are serious about creating anything for iOS then buy a mac.

First off, you did notice:

make mobile apps for iOS and Android

Right?

Then, did you notice the very first response?

Then I am assuming you didn't notice the OP's clarification where he stated he didn't have a budget, this taking Construct2, Unity or GameMaker off the table?

So yeah, people in glass houses and all of that...