I think that many times begin and start can be used interchangeably.
But one difference is that when you are turning on a peice of machinery (car, computer), you can say, "I am starting the car." You can't use begin for that.
Also, these is what wordweb says:
begin:verb: begin (began, begun, beginning)1) Take the first step or steps in carrying out an action
"We began working at dawn"; "The first tourists began to arrive in Cambodia"; "He began early in the day."2) Have a beginning, in a temporal, spatial, or evaluative sense
"The DMZ begins right over the hill"; "The second movement begins after the Allegro"3)Set in motion, cause to start
"The Iraqis began hostilities"; "begin a new chapter in your life"4) Begin to speak or say
"'Now listen, friends', he began"5) Be the first item or point, constitute the beginning or start, come first in a series
"The number 'one' begins the sequence"; "A terrible murder begins the novel"; "The convocation ceremony officially begins the semester"6) Have a beginning, of a temporal event
"WWII began in 1939 when Hitler marched into Poland"; "The company's Asia tour begins next month"7) Have a beginning characterized in some specified way
"The novel begins with a murder"; "My property begins with the three maple trees"; "Her day begins with a work-out"; "The semester begins with a convocation ceremony"8) Begin an event that is implied and limited by the nature or inherent function of the direct object
"begin a cigar"9) Achieve or accomplish in the least degree, usually used in the negative
"This economic measure doesn't even begin to deal with the problem of inflation"; "You cannot even begin to understand the problem we had to deal with during the war"10) Begin to speak, understand, read, and write a language
"She began Russian at an early age"
Noun: Begin
1) Israeli statesman (born in Russia) who (as prime minister of Israel) negotiated a peace treaty with Anwar Sadat (then the president of Egypt) (1913-1992)
start:
Noun: start
1)The beginning of anything
"it was off to a good start"
2)The time at which something is supposed to begin
"they got an early start"
3)A turn to be a starter (in a game at the beginning)
"he got his start because one of the regular pitchers was in the hospital"
4)A sudden involuntary movement
"he awoke with a start"
5)The act of starting something
6)A line indicating the location of the start of a race or a game
7)A signal to begin (as in a race)
8)Advantage gained by an early start as in a race
"with an hour's start he will be hard to catch"
Verb: start
1)Take the first step or steps in carrying out an action
"Who will start?"
2)Set in motion, cause to start
"The U.S. started a war in the Middle East"
3)leave
4)Have a beginning, in a temporal, spatial, or evaluative sense
"Prices for these homes start at $250,000"
5)Get off the ground
"Who started this company?"; "I start my day with a good breakfast"; "The blood shed started when the partisans launched a surprise attack"
6)Bring into being
"Start a foundation"
7)Get going or set in motion
"We simply could not start the engine"; "start up the computer"
8)Move or jump suddenly, as if in surprise or alarm
9)Begin or set in motion
"I start at eight in the morning"
10)Begin work or acting in a certain capacity, office or job
"start a new job"
11)Play in the starting line-up
12)Have a beginning characterized in some specified way
13)Begin an event that is implied and limited by the nature or inherent function of the direct object
"She started the soup while it was still hot"; "We started physics in 10th grade"