C++ started out as basically C with extra features (most notably the 'class') to support object-oriented programming. So most of the basic constructs are the same, or very similar, in both languages: arithmetic expressions (calculations), boolean expressions (comparisons, etc.), if-statements, loops, ordinary functions, etc.
Where the two languages differ is basically that (a) C++ has features that directly support object-oriented programming, whereas C does not, and (b) C++ has a more extensive standard library than C does, for performing common tasks.
At the very beginning level, the biggest difference is in the way you normally do input and output. For example, C does output using the printf() function and its relatives, whereas C++ usually uses iostreams (e.g. 'cout' for standard output), which are easier to use for simple input and output. However, C++ also allows you to use C-style output via printf(), for backwards compatibility with C.
Going a bit further, both C and C++ let you group data together using something called an "array", which works pretty much the same way in both languages. C++ also has something called a "vector" which can do things that an array cannot and is much more convenient to use than an array in many situations.
Similarly, in both C and C++ you can store text as an array of characters, but C++ also has a 'string' data type that is specifically designed for that purpose, which is more convenient and less prone to programmer errors.
Many C++ books, especially older ones, look a lot like C books, because they're basically C books that were re-tooled into C++ books. The language is C++, but they do things in a C-style way, using the older features inherited from C rather than the newer ones that were created for C++.
For an example of a "modern" approach to C++, try Koenig and Moo's "Accelerated C++".
At the college level, I think you'll find that most places use Java for their introductory programming courses now, because of its ties to the Web and because it has an even more extensive standard library than C++ does. The Java standard library includes graphics, networking, etc., which the C++ standard library does not. You can still do all that stuff in C++ or C, but you have to use libraries that are tailored for your operating system: Unix versus Windows versus MacOS, whereas in Java it's the same everywhere (at least in principle).