The reason for replacing Rodgers with Klopp was for a change to improve results. It's very early but we can look for some changes in what Klopp is looking towards in comparison to what Rodgers was looking towards.
Rodgers came in with a preferred formation of 433, with the midfield 3 being his 2-1 (looks a lot like a 4231) or 1-2 (the present 433). He wanted "death by football" which was a combination of high pressing on defence and possession in attack.
Klopp comes in with a preferred formation of 433/4231 similar to Rodgers, with high pressing in defence but an ideal of fast transitions when in attack.
Klopp has come in and instituted his pressing defensive style right away, and the defensive results have been good, 2 goals in 3 games is excellent. However, we saw Rodgers use a pressing style to great effect, noticeably against Arsenal. However, the intensity of Rodgers' pressing was lower, lasted for fewer minutes in a game, and was a far cry from appearing in every game. It seems to me that the intensity of the pressing under Klopp has already decreased markedly from that first game against Tottenham, which will happen when you play games every three days with basically the same line-up. What is the big difference between what Rodgers wanted and what Klopp wants? What is the difference in problems they face getting what they want?
The basic concept of the attacking side of the game is markedly different, patient, probing possession against immediate attack (put in the most extreme manner.) Rodgers wanted to starve the opposition of the ball, percentage of possession mattered to him. Klopp is completely happy with possession flipping rapidly back and forward as long as that happens in the other team's half, chaos in their half, structure in ours. However, what was the style that Rodgers had the most success using? Rapid, incisive counter-attacks whenever the ball was turned over, wherever that was on the field. Rodgers attacking style this year seemed, to me, closer to that than a pure possession-based game.
It's very early, and I am far from completely up on Klopp's tactics, but I wonder what are the real differences between the two? Klopp has been far more successful in the past, but doing precisely what that was different from what Rodgers was doing in the last two and a half years? The most striking thing about the new Klopp era is how close it looks to the old one, other than the players working a lot harder for the new coach.
Strikers get most of the plaudits, but it might actually be true that they are under-estimated in value.
My wife is a Newcastle fan, and under Pardew they had some of the largest swings in performance I have ever seen. It was as clear as night and day to me that these swings were based around whether he had a striker who could score. Demba Ba scored at an amazing rate for half a season, and then Papiss Cisse was ridiculous for the second half of the season, and they finished fifth. Ba left and Cisse got injured (and returned to earth) and they finished 16th. Loic Remy was a reliable finisher, but when he left they collapsed again. Demba Ba versus Shola Ameobi is worth multiple places in the table.
Last year it was Sturridge versus Balotelli etc., worth how many places? One? Two? Two is 4th and Rodgers still has his job. Benteke and Sturridge is worth how many places?
This year it is Sturridge and/or Benteke against Ings (at first) and now Origi. It's so clear how much better we are in attack with either Sturridge or Benteke to anyone watching. Even half a goal a game difference would be enough for two of our five draws to be wins, and us ahead of Spurs and the favourites for fourth, and Rodgers still with a job.
There are some differences I can see. Klopp clearly is a more defensively-oriented coach, and we already look more organized. I don't assume a goal will come from each opposition's set piece (although the results don't seem different yet). Klopp is a more likable figure, will be more popular with players and fans, and possibly helpful with recruiting. Oddly enough Klopp seems more committed to his style than Rodgers, but that style seems very similar to what Rodgers was actually using.
I'm not opposed to Klopp at all, but I thought Rodgers was a good coach. I think Klopp is probably better, mostly based on his defensive abilities, Klopp has a track record that instills belief, and people were burnt out on Rodgers' schtick. However, I think a much bigger deal is the health of our two, very good, strikers. Sturridge or Benteke playing over Origi is a bigger difference than Rodgers or Klopp managing. The two of them playing together is multiple spots in the table.
What I expect for this season is something that over time looks a lot like us under Rodgers. Over the next couple of months things will seem bitty, the intensity will dwindle as our main players tire. Then at some point Sturridge and/or Benteke will get healthy, the team will get familiar with each other and the coach, and we will go on an exciting run. If we are close enough to legitimately challenge for fourth then the team will run and run for Klopp, and if they aren't the intensity level will drop and we'll finish somewhere between 5th and 7th.
That's a lot to take from only three games, but that's what internet forums are for.