Let me give a few sample cases where you want one and not the other.

If you want to use the return value, then you want to use do: 

# Trick for generating a symbol to use as, for instance,
# a filehandle...
my $fh = do {local *FH};

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If you want to use loop control, then you have to use a bare block. merlyn gives some ideas at RE: My favorite looping mechanism in Perl is: about why you might want to do that.

If you want to put several statements where syntactically only one is supposed to go, you would want do. That might look like this: 

do {print "Hello"; print " World\n";} for 1..10;

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Note that I don't feel the need for this use of do. So I will follow up with a use for bare blocks that I feel is equally useless. Suppose you don't like semi-colons. Well help is at hand! 
{print "Hello"}
{print " World\n"}
# etc

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Does that clarify the similarities and differences?

In reply to Re (tilly) 1: {} vs do{} by tilly
in thread {} vs do{} by John M. Dlugosz

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