Both Smoke & Sanity testing are same in one way that is, 
its a build verification testing. When a particular module 
is developed then developer will do smoke testing in order 
to test whether the developed module is stable (which means 
the basic functionalities must work) then in turn it will 
be given for testing.

Now the testers will also do the build verification testing 
where we call that as Sanity testing

smoke testing is a non-exhaustive testing where as sanity 
testing is a cursory testing,This level of testing is a 
subset of regression testing . So I dont think smoke 
testing is also called sanity testing

its not right to say that smoke testing and sanity testing 
is same. both are different in some context. sanity testing 
comes first and then smoke testing. sanity testing is the 
initial level of testing required if the functionality 
which is being tested passes through this test then it will 
go for the next level of testing.
THE BASIC DIFFERENCE BETWEEN THEM ARE :
(1)
Smoke testing originated in the hardware testing practice 
of turning on a new piece of hardware for the first time 
and considering it a success if it does not catch fire and 
smoke.  In software industry, smoke testing is a shallow 
and wide approach whereby all areas of the application 
without getting into too deep, is tested. 
 
WHERE AS 

A sanity test is a narrow regression test that focuses on 
one or a few areas of functionality. Sanity testing is 
usually narrow and deep.

(2)
A smoke test is scripted--either using a written set of 
tests or an automated test

WHERE AS 

A sanity test is usually unscripted

(3)
A Smoke test is designed to touch every part of the 
application in a cursory way. It's is shallow and wide.

WHERE AS 

A Sanity test is used to determine a small section of the 
application is still working after a minor change.

(4)
Smoke testing will be conducted to ensure whether the most 
crucial functions of a program work, but not bothering with 
finer details. (Such as build verification). 

WHERE AS 

Sanity testing is a cursory testing; it is performed 
whenever a cursory testing is sufficient to prove the 
application is functioning according to specifications. 
This level of testing is a subset of regression testing.

SO THESE ARE THE GENERAL DIFFERENCE

I do agree with the above statement where nishi explained
but one this is wrong according statement.

Smoke testing is done first then comes the sanity testing.

Smoke teting is done by developers to check the basic code
part and there functionality is working fine or not before
giving the build to developers.

When it comes to testers they will do a sample round of
testing where we will be checking the basic functionality of
testing and give a clearance that this application meets the
minimum functionality to carry forward for other functional
testing.

rest of other things which nishi explained is right.


Always Smoke testing comes first then sanity testing.