On responding with 13 pts and 4-card major support why it is important to make a delayed raise (e.g. change suit and only then jump to 4-lvl) and not jump directly to 4-lvl- what added value does it supply to the partnership? What would go wrong if I'd just jump to 4-lvl - would our partnership be deprived in some way?
A common bridge principle for SAYC is to bid up strong hands slowly, but jump to a high bid quickly with weak hands that have long suits or other preemptive features.
In the situation you describe with responder having 13 points, you'll want bidding space to communicate whether or not you should be in game or slam, because opener has promised 13-21 points (and therefore the combined hands have 26-34 points).
If slam is possible, most tables will bid it up slowly and find the right slam contract, while your hasty 4 bid lands you a mere game contract (which will give you the bottom score in duplicate bridge).
Therefore, in SAYC, the jump to 4 of the opener's major response is reserved for a weak bid that typically indicates a small number of points but good shape.
For example, with 6 low trump, a void, and a single Ace for high card points, that would be a perfect hand to jump to the 4 level because:
The key for an immediate jump to 4 is to be sure that slam is (nearly) impossible opposite partner's strong hand, while providing a good play for game opposite partner's expected minimum opening. Hands such as these qualify over 1 Spade by partner:
None of these offer a decent slam opposite
despite the massive fit because, among other reason, they don't have an ace, a void, or two kings. However they still offer a play for game opposite minimum hands such as these:
If you jump from say, one spade to thtree spades (four spades in hand) or four spades (five spades in hand), you are saying that you have spades, and nothing else, with most of your 6-9 points concentrated in the trump suit. You might just make your contract, but even if you go down, the idea is that you will have kept your opponents out of a two-or three-level contract that would have scored higher than the penalty you suffered.
By using a delayed raiese, you show values in a "second" suit that offers an alternative to the bid suit, that could offer the possibility of a game in the second suit, or no trump. Let's say the bidding is one spade, two hearts, with your bid showing four hearts and three spades. If partner has four hearts, he might bid four hearts with a 4-4 fit and use the 5-3 spades as a side suit. If his distribution is 5-2-3-3, he might bid no trump.
Also, your delayed raise would show 10+ points. Opposite an opening hand, you probably have game if you can find a suitable fit; the bid of a "new" suit makes it easier to find the best fit.