What is the difference between indistinguishable particles and identical particles?
If I create an electron on earth and someone else creates an electron on Andromeda, they're identical particles. They have the same quantum numbers, they're both excitations of the electron field. However they're distinguishable by means of their spatial separation. Their wavefunctions don't overlap.
Edit: perhaps I should add that not everyone uses the two words in this strict sense. Sometimes they're used interchangeably, but blurring them carries with it the danger of taking seriously the entanglement implied by antisymmetrizing across all existing electrons.
Two particles are the same if you can't tell the difference, that is if the exchanging operator commutes with all observables. In this QM context thus indistinguishable and identical particles mean the same thing.
This is one of those cases where language problems make quantum theory nightmare to learn. In daily life, two things are identical when it is hard or impossible to tell the difference in their properties by casual examination. For example, two xerox copies of the same document, two balls from the same metal bearing etc. This does not mean that they are absolutely the same. In daily life there is no such pair, but we want to use the word identical.
Indistinguishable in daily life means pretty much the same thing.
In case of electrons, we call them identical because they have the same charge and mass and possibly other parameters (spin...) for similar reasons as we call two xerox copies identical.
But for electrons the word indistinguishable is often used in a different sense than the above daily sense. What people often mean is that in the atom, one cannot follow electrons in time to maintain the knowledge of which is which. In this sense, in classical physics identical particles are distinguishable, but in quantum theory identical particles are indistinguishable, because we cannot look and track the electron in the atom.
What is of greater importance, however, is that it was found that description of two or more electrons in an atom is in best agreement with measurements when anti-symmetric Hamiltonian eigenfunction is used. This is such that no difference between the properties of any two different electrons can be derived from it. That is also one meaning of "electrons are indistinguishable". If they were described by function that is neither symmetric nor anti-symmetric, predictions for one electron would be different than for the other and the electrons would be distinguishable in this sense. But in atom only anti-symmetric functions seem to be appropriate.
For large enough systems or separations, all this loses validity. For example, the electron in your fingernail is of course distinguishable from the electron in my tooth. Anti-symmetric wave functions are inappropriate for such cases.
Sometimes, by indistinguishable people mean "the two electrons do not have individuality", "interchange of two electrons is not a real event" or even "all the electrons are just one electron" or similar weird stuff, but there is little evidence for that in physics.