This question is a good first stop for students of the opening.
Comparing the two:
rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR w KQkq e3 0 1
Less influence over the centre: 1.e4 doesn't stop the freeing pawn moves 1...c5 (Sicilian Defence), 1...d5 (except at top level, Scandinavian Defence), 1...e5 or 1...f5 (was Black going to weaken his king with this move anyway?). No control over the d4 or e4 squares.
Less coordination: White's e4 pawn is left floating unsupported in the middle of the board.
More active pieces: White's queen and bishop freed for operations on the wings.
Better king safety: 1.e4 facilitates rapid castling after 2.Nf3 ... 3.Bc4, for example.
rnbqkbnr/pppppppp/8/8/3P4/8/PPP1PPPP/RNBQKBNR w KQkq e3 0 1
More influence over the centre: 1.d4 stops the freeing pawn moves 1...c5 and 1...e5, and controls the d4 square via the queen.
More coordination: White's d4 pawn is supported by the queen.
Less active pieces: White's queen and kingside (castling) bishop development hindered compared to 1.e4, but an eventual fianchetto of the kings bishop to the g2 square, although slower, may be more active and point more effectively toward the centre in the long run.
Less king safety: White's kingside pieces will take longer to develope hindering the preferred kingside castling. The King is opened to immediate bishop-checks along the e1-a5 diagonal (Nimzo-Indian Defence, for example)
So, in very broad general terms, with 1.e4 White is wanting to throw his king into a castle then charge forth into the fray; whereas with 1.d4 White takes a more controlled approach, on his opponent's forces, on his own forces via piece coordination, and is confident enough to handle his king dallying in the centre.
If you accept all of the above, we can then use this knowledge to make decisions about our opening repertoire and move decisions in the opening. For example, some questions might come to mind given all of the above, like:
- As White with 1.d4, what is the line that allows the fastest castling so that I can blunt one of 1.d4 openings weaknesses while retaining its advantages?
- As Black facing 1.d4, what is a reply that takes advantage of White's slow castling, perhaps something that involves a quick attack that will catch his king in the centre?
- As Black facing 1.e4, what is a reply that results in all my pieces set up to attack White's kingside as that is where his king is likely to end up.
Etc. There are many questions that suggest themself after comparing the pros and cons of 1.e4 vs 1.d4. It is true that perhaps making any conclusions is premature based on just one move, but then your opponent might make follow-up moves that exacerbate one of the above stated disadvantages further...
And then there are other considerations unrelated to the disposition of the pieces on the board:
- The sum total of the theory in existence on 1.e4 is likely more than there is for 1.d4 openings. The Sicilian Defence in particular is a vast body of theory, but also the centuries old Ruy Lopez, the French, etc. You may be inclined to avoid this theory.
- Former World Champion, and one of the most dominant players in chess history, Bobby Fischer preferred 1.e4 and used it almost exclusively throughout his career, describing it as "Best by test."
There are differences. I encourage any chess player to really look closely at the two before deciding, and base their decision on their own critical analysis rather than just blindly accepting known maxims like "beginners should play 1.e4 because it results in open positions and tactical games."