There are two basic differences. One is electronic and has to be sent and received by a person trained in Morse Code; a letter is a physical document you create and send through the postal service. The other is the cost. If you send a telegram, you pay by the word, so telegrams tend to be cryptic and much shorter than letters.
The rise of e-mail has provided a method of communication that has the best of each of these ... it's almost instantaneous, and there is no direct charge for each message.
I have never sent a telegram and I don't know anyone who has.
Letters are written documents that can be sent by literally anyone; all they necessitate is writing something down on paper, putting it in an envelope, addressing it to the desired recipient's address, applying the appropriate number of stamps, and dropping it off at the post office. Telegrams, on the other hand, are messages which are sent by wire or broadcast signals through a telegraph machine. They were--at least at the time during which the novel is set--much quicker to send than letters. You can read more about telegrams at the BBC link below.
A telegram appears in Chapter Twelve of Christopher Paul Curtis' Bud, Not Buddy. After Bud hitches a ride to Grand Rapids with Mr. "Lefty" Lewis, Bud falls asleep. When he wakes up, he discovers two things: that the box of blood has been dropped off at the hospital and that Lefty has sent a telegram to Bud's "father" to warn him that Bud will be returning home shortly. This is good cause for concern for Bud because the story he told about leaving home is a lie.