I'm going to first assign variables.

Bn = Bobby now
Jn = Johnny now
Bb = Bobby before
Jb = Johnny before

Then, from the prompt, I can write the following equations.

Jn = Bb
Jb = 1/2 * Bn

Statement 1:
I can write the following equations.

Bn = 4*(Bn - Jn) since (Bn - Jn) is the age difference between Bobby and Johnny
Bn = 4*(Bb - Jb)

Now, I have four equations and 4 unknowns. However, I still can't find a unique solution. I see, just by looking at the equations, that they can be solved by setting Bn = Bb = Jn = Jb = 0. And, Statement 2 tells me that it's possible for Bobby and Johnny to have different ages, so Statement 1 is insufficient.

Statement 2:
Again, I can write 2 new equations.

Bn = Jn + 6
Bb = Jb + 6

And again, I have four equations, but this time, there is a constant (6) so I can't solve them by setting them all equal to 0. So 4 equations and 4 unknowns: Statement 2 is sufficient.

Bn = 24, Jn = 18, Bb = 18, Jn = 12

Another simple solution:
Let Bobby's age now=B and Johnny's age now=J years. So the diff in their ages is=B-J

So, Johnny was half old as Bobby is now=B/2
Then, Bobby's age was=B/2+(B-J)
As per stem,
J=B/2+(B-J)
or, 2J=3B/2
or, 4J=3B

Statement 1: When Johnny was born Bobby's age was B-J
So, B=4(B-J)
or, 3B=4J
Insufficient.
Statement 2: B-J=6

So now we have 2 equation,
B-J=6 and
4J=3B
or 3B=4(B-6)
or B=24
Sufficient.