I hope I am not going to get into trouble for posting a piece of a free newsletter by GM Alex Colvic on the subject you asked about. This is the entire email from April 28,2018. Alex's answers the emails at the address email@example.com
The main advantage of the Petroff over the Berlin is that it is more practical. The Petroff arises already on move 2, thus avoiding very important White options after 2...Nc6 such as the Scotch, the Italian and also the Ruy Lopez with 4 d3.
The return of the Petroff is of course thanks to Fabiano Caruana's success with it at the Berlin Candidates. Usually considered a boring opening aimed only at playing for a draw, Caruana actually managed to win quite a few games with it: two at the Candidates (against Kramnik and Grischuk) and also one in the Grenke Classic against Vitiugov.
Another positive aspect of the Petroff is that there are only a few critical lines against it. In what has traditionally been called the main line, after 3 Ne5 d6 4 Nf3 Ne4 5 d4 d5 6 Bd3, Caruana's choice has been the move 6...Bd6, instead of the previously considered more solid 6...Nc6. The reason for his choice is that after 6...Nc6 7 0-0 Be7 8 Nbd2, a move previously considered harmless, manages to pose unpleasant problems to Black in a symmetrical position.
Worth noting here is that the Chinese players have their own way of treating this line. They introduced the move-order 6...Bf5 7 0-0 Be7 (or 6...Be7 7 0-0 Bf5). One of the ideas of this move-order is to be able to play ...Nd6, exchanging the light-squared bishops.
Caruana demonstrated that Black is theoretically OK in the main line with 6...Bd6 and that is probably the way to go.
The second critical attempt is 3 Ne5 d6 4 Nf3 Ne4 5 Nc3. Here taking on c3 is most popular, though there is something to be said for the retreat 5...Nf6, when White doesn't have the active plan with c4 at his disposal. After 5...Nc3 6 dc Be7 7 Be3 Nc6 Black usually castles queenside and should be fine. But this position is deceptively simple, as White can find small ways to keep the pressure on, demonstrated by Caruana's only loss in Berlin to Karjakin. Black has also plans with short castling in this line, as Hou Yifan played against Caruana himself and almost beat him.
The third attempt is to play 3 d4. Now both moves, 3...ed and 3...Ne4 are possible. Caruana's choice was the latter, and after 4 Bd3 (Grischuk and Vitiugov played 4 de in the above-mentioned games) d5 5 Ne5 (5 de is another possibility) Nd7 Black again is theoretically fine.
The interesting part with the Petroff now is that it comes back after a certain period of being not very popular. This means that the old lines and recommendations are ripe for re-evaluation. The improved engines will suggest new moves (check out 5...Qd7 in the game Vitiugov-Caruana) and new ideas and variations will appear. This is the usual process when an opening "comes back" after some time spent on the sidelines.
This is how modern theory evolves. The latest engines start suggesting moves and ideas and they inject new life in the old openings, once considered boring or harmless.
Perhaps it is time to take up the Petroff!
Have a great weekend,