Playing the ponziani is a trap.
Ponziani opening traps

Playing the ponziani is a trap.
That's true. Black has better chances than white if he plays 3... d5 but I almost never see that line, so it doesn't worry me.

Yes, the best line for Black is 3. ... d5 , where the only answer is 4. Qa4 . It can be very tough for White.
For a similar Ponziani trap I usually play, succesfully, (see http://scacchi.vecchilibri.eu/partite/ponziani.html )
1. e4 e5 2. Nf3 d6 3. c3 Nc6 4. d4 Nf6 5. Be2 {waiting move } Nxe4 {? takes the bait } 6. d5 Ne7 7. Qa4+ {! double threat } c6 8. Qxe4
one piece up

4...Nxe4 isn't a horrible move by black, it is just a different approach which is played most of the times

The Ponziani is very tricky. When I played on the Ponziani Power team we were very often winning by the 13th move.
There are thousands of ways Black can go wrong. Here is one of them from a game I played in 15 minutes per side chess about 4 years ago:

I actually considered playing the Ponziani recently in an OTB game, but was put off by the line 3...d5 4.Qa4 Bd7.

sammynoury, please explain what happens if 4..exd4.
You get something like: 4... exd4 5. e5 Ne4 6. Qe2 f5 7. exf6
d5 8. Nbd2 Qxf6 9. Nxe4 dxe4 10. Qxe4+ =
There are a bunch of things your opponent could do wrong while playing the ponziani, which is almost a trap opening. Which ones do you know, would be great to know them all. Here are two of the most infamous, pretty similar though.