Some people use these terms interchangeably. There is a difference though and the financial consequences between them are significant.
An altered stamp is something that is genuine, but it has been changed, most times to increase its value. A cancellation has been removed. The stamp was reperforated to remove a straight edge. The stamp has been regummed to mask a fault or make it appear as never hinged.
At its core level though, the altered stamp is a genuine stamp.
A fake is something that it is not supposed to be. Unlike an altered stamp, a fake is not genuine at its core. For example, someone takes an imperforate Washington-Franklin head issue and adds some perforations to make it appear to be a rare coil issue. In this case, at its core level, this stamp is not supposed to be a coil.
Fakes and altered stamps do share one aspect though – which is probably why collectors interchange the terms. Sometimes, these stamps increase significantly in value. Take a mint copy of Scott #245 ($5 Columbian) that has no gum and add gum to it to make it appear as if it is never hinged (this is an altered stamp). The value is going to skyrocket. Conversely, take a wide margin sheet stamp and trim away the side perforations to make it look like a rare coil issue (this is a fake). It too has gone way up in value.
Reprints are done using the original stamp design. For example, take Scott #10X1 and #10X2. These are the Providence, RI Postmaster Provisionals. Mint copies of the originals are uncommon, but by no means rare.
In 1898, the stamp dealer firm of Bogert & Durbin was able to make reprints of these postmaster provisionals from the original printing plates. The reprints all have a letter on the reverse side from the firm name: “B.O.G.E.R.T.D.U.R.B.I.N.” Wisely, they stamped the letter on the back of the stamp to mark them as reprints so that they are not confused with the original stamps. The reprints are pretty common and even full sheets of 12 of the reprints are known.
Here is the key. Reprints are made to look exactly like (or as near as possible) to the original.
Reproductions, on the other hand, are not made from the originals. And in most cases, reproductions should not fool anyone. The design has changed. They are printed in different colors of ink than the originals. And so on.
Take the Confederate stamps. Because the Confederacy is dead and their stamps have no postal validity, some stamp companies went to work creating reproductions of these stamps. Because they were not usable as postage, no one from the US Treasury Department or the Secret Service was going to arrest them for creating these stamps. The reproductions are usually imperforate and many times similar to, but not identical to the original stamps. Often times, they were printed in different colors too.
These reproductions sold for less than a dollar or two. Sometimes you could find these stamp reproductions in a gift shop that was selling other souvenir items.
Technically speaking, Scott #3 and #4 are reproductions too. In this case, they are not reprints since they didn’t come from the original printing plates that made Scott #1 and #2. However, these reproductions are very similar to the real McCoy’s. There are subtle differences in the designs. To an uneducated collector, they could easily be mistaken for Scott #1 and #2. For someone who knows of their existence, detecting these reproductions is not difficult.
Reproductions are not meant to be dangerous. They could never be sold to someone else as original items unless the buyer was totally unaware of what the originals look like.
And lastly, counterfeits. Yes, they are fake stamps. But there is a twist. Counterfeits though are meant to fool a postal administration into thinking they are legitimate stamps that are paying postage. Fakes are usually targeted at collectors.
Obviously, counterfeits have to look almost exactly like the real McCoy or their appearance on envelopes is going to at least raise suspicions of postal officials.
Fakes tend to be a single stamp or a small group of stamps at the most. However, counterfeits are usually produced in large quantities: by the thousands, or more. Counterfeiting a stamp takes a lot of time to make sure the appearance is right, the same paper is used, and so forth. For the hours involved, no one is going to just create a few counterfeits. They will create as many counterfeits as possible and hopefully sell them below face value by the hundreds or thousands to unsuspecting businesses looking to curb their postage costs a little bit.