Is Arduino IDE enough to program with C or C++?
In terms of the core language features, there's virtually no difference. In the background, the Arduino IDE and Atmel/AVR Studio both use GCC to compile the code.

GCC is an excellent toolset which supports all the core language features of C++. Admittedly, the version which currently comes with the Arduino IDE is a little old, so it doesn't support C++11 onwards. However, a lot of C++11 features are not very relevant or useful for microcontroller programming anyway (e.g. threading).

You're also likely to use the same standard library for both, which is AVR Libc. It provides pretty much all the standard C functionality that you're likely to use in an AVR project. It doesn't include the C++ stuff though, such as streams, containers, and algorithms. These tend to be very resource-hungry though, so are best avoided anyway.

The major difference is basically flexibility. Atmel/AVR Studio is designed to let you work on a wide range of projects and customize the build to suit very specific needs. You can also use alternative libraries with it more easily. In theory, the Arduino IDE can be modified in similar ways, but it's a major hassle because it provides no GUI to do it.

It's also important to note that the Arduino IDE does a number of non-standard things, such as re-organising declarations and including libraries in strange ways. It's designed to be more user-friendly, but often ends up just getting in the way. There are usually ways to work around those issues though.