gcc being a compiler/linker, its -s option is something done while linking. It's also not configurable - it has a set of information which it removes, no more no less.
strip is something which can be run on an object file which is already compiled. It also has a variety of command-line options which you can use to configure which information will be removed. For example, -g strips only the debug information which gcc -g adds.
Note that strip is not a bash command, though you may be running it from a bash shell. It is a command totally separate from bash, part of the GNU binary utilities suite.
The accepted answer is very good but just to complement your further questions (and also as reference for anyone that end up here).
What's the equivalent to gcc -s in terms of strip with some of its options?
They both do the same thing, removing the symbols table completely. However, as @JimLewis pointed out strip allows finer control. For example, in a relocatable object, strip --strip-unneeded won't remove its global symbols. However, strip or strip --strip-all would remove the complete symbols table.
Which one do you use to reduce the size of executable and speed up its running
The symbols table is a non-allocable section of the binary. This means that it never gets loaded in RAM memory. It stores information that can be useful for debugging purporses, for instance, to print out a stacktrace when a crash happens. A case where it could make sense to remove the symbols table would be a scenario where you have serious constraints of storage capacity (in that regard, gcc -Os -s or make CXXFLAGS="-Os -s" ... is useful as it will result in a smaller slower binary that is also stripped to reduce size further). I don't think removing the symbols table would result into a speed gain for the reasons commented.
Lastly, I recommend this link about stripping shared objects: http://www.technovelty.org/linux/stripping-shared-libraries.html
From the (Mac OS X, but others are similar) man page:
strip removes or modifies the symbol table attached to the output of the assembler and link editor. This is useful to save space after a program has been debugged and to limit dynamically bound symbols.
Note the bit about "after a program has been debugged" because debugging a stripped executable is very hard, indeed. The "limit dynamically bound symbols" is a rarer use: it lets you make some of the functions in an external library inaccessible by taking away the index entries that tell where the actual code is located. This is also explained in the man page.
As cheap and plentiful as disk is in most situation you simply wouldn't bother with this anymore. But you might want it for space constrained situation like embeded devices, rescue disks, etc.
It strips symbol information from the binary. The binary contains some information that maps symbols (e.g. function names) to particular locations. strip removes those.