How different is Objective-C from C++?

Short list of some of the major differences:

• C++ allows multiple inheritance, Objective-C doesn't.
• Unlike C++, Objective-C allows method parameters to be named and the method signature includes only the names and types of the parameters and return type (see bbum's and Chuck's comments below). In comparison, a C++ member function signature contains the function name as well as just the types of the parameters/return (without their names).
• C++ uses bool, true and false, Objective-C uses BOOL, YES and NO.
• C++ uses void* and nullptr, Objective-C prefers id and nil.
• Objective-C uses "selectors" (which have type SEL) as an approximate equivalent to function pointers.
• Objective-C uses a messaging paradigm (a la Smalltalk) where you can send "messages" to objects through methods/selectors.
• Objective-C will happily let you send a message to nil, unlike C++ which will crash if you try to call a member function of nullptr
• Objective-C allows for dynamic dispatch, allowing the class responding to a message to be determined at runtime, unlike C++ where the object a method is invoked upon must be known at compile time (see wilhelmtell's comment below). This is related to the previous point.
• Objective-C allows autogeneration of accessors for member variables using "properties".
• Objective-C allows assigning to self, and allows class initialisers (similar to constructors) to return a completely different class if desired. Contrast to C++, where if you create a new instance of a class (either implicitly on the stack, or explicitly through new) it is guaranteed to be of the type you originally specified.
• Similarly, in Objective-C other classes may also dynamically alter a target class at runtime to intercept method calls.
• Objective-C lacks the namespace feature of C++.
• Objective-C lacks an equivalent to C++ references.
• Objective-C lacks templates, preferring (for example) to instead allow weak typing in containers.
• Objective-C doesn't allow implicit method overloading, but C++ does. That is, in C++ int foo (void) and int foo (int) define an implicit overload of the method foo, but to achieve the same in Objective-C requires the explicit overloads - (int) foo and - (int) foo:(int) intParam. This is due to Objective-C's named parameters being functionally equivalent to C++'s name mangling.
• Objective-C will happily allow a method and a variable to share the same name, unlike C++ which will typically have fits. I imagine this is something to do with Objective-C using selectors instead of function pointers, and thus method names not actually having a "value".
• Objective-C doesn't allow objects to be created on the stack - all objects must be allocated from the heap (either explicitly with an alloc message, or implicitly in an appropriate factory method).
• Like C++, Objective-C has both structs and classes. However, where in C++ they are treated as almost exactly the same, in Objective-C they are treated wildly differently - you can create structs on the stack, for instance.

In my opinion, probably the biggest difference is the syntax. You can achieve essentially the same things in either language, but in my opinion the C++ syntax is simpler while some of Objective-C's features make certain tasks (such as GUI design) easier thanks to dynamic dispatch.

Probably plenty of other things too that I've missed, I'll update with any other things I think of. Other than that, can highly recommend the guide LiraNuna pointed you to. Incidentally, another site of interest might be this.

I should also point out that I'm just starting learning Objective-C myself, and as such a lot of the above may not quite be correct or complete - I apologise if that's the case, and welcome suggestions for improvement.

EDIT: updated to address the points raised in the following comments, added a few more items to the list.

While they are both rooted in C, they are two completely different languages.

A major difference is that Objective-C is focused on runtime-decisions for dispatching and heavily depends on its runtime library to handle inheritance and polymorphism, while in C++ the focus usually lies on static, compile time, decisions.

Regarding libraries, you can use plain C libraries in both languages - but their native libraries are completely different.

Of interest though is that you can mix both languages (with some limitations). The result is called Objective-C++.

Off the top of my head:

1. Styles - Obj-C is dynamic, C++ is typically static
2. Although they are both OOP, I'm certain the solutions would be different.
3. Different object model (C++ is restricted by its compile-time type system).

To me, the biggest difference is the model system. Obj-C lets you do messaging and introspection, but C++ has the ever-so-powerful templates.

Each have their strengths.