P vs. NP, technically vs. morally
As <PERSON> said it is possible that P=NP is technically true but morally false. P=NP is morally true (even if not necessarily technically) if there is a fast deterministic algorithm (say $O(n^2)$, or maybe even $O(n^{\lg^* n})$ with small constants, nothing like $2^{65536}+2^{1024}n^{256}$) that solves one of famous NP-complete problems like SAT. IIRC, <PERSON> once said that he would consider the P vs. NP problem as essentially settled if someone shows SAT can be solved in $O(n^{\lg n})$ time.
So what happens if P=NP is morally true?
This brings us back to why NP is such an interesting class of problems. The intuition in general is that we often want to find objects of reasonable size (formalized as polynomial size) that hold a property $Q$ and the property $Q$ is easy to verify (formalized as computable in polynomial time). This class of problems encompasses almost all problems we are interested in. To go beyond it you need to think about interactions between players like games. The number of natural interesting problems that are not in NP (or PH) is very small compared to natural interesting NP problems. 603
If P=NP is morally true then all of these problems can be solved very fast. Just to give an example, you can learn best weights for very complicated machine learning models. You can break encryption protocols.
Comparison with the case where P$\neq$NP is morally true
By P$\neq$NP is morally true I mean that we cannot solve SAT (or any of other famous NP-complete problems) much faster than brute-force then these problems cannot be solved in practice for general inputs of even rather small input size of say 100.
Does P$\neq$NP is morally true mean we cannot solve NP-hard problems in practice?
Even if P$\neq$NP is morally true, it is still possible that for some of these problems we are interested not in general inputs and worst-case but a class/distribution of inputs that can be solved efficiently. E.g. it can be the case that solving SAT in the worth case requires exponential time but in practice we can already solve SAT on many interesting classes like software verification, hardware verification, etc. much faster.
This is kind of similar to solving a simpler problem, e.g. TSP cannot be even approximated efficiently if P$\neq$NP is morally true, yet we already can approximate the special case of TSP on Euclidean graphs.
If you know you want to solve an NP-complete problem not on general inputs but on inputs with particular properties you don't need to care about the general problem. You only need to solve the simpler problem. Unfortunately it is often not easy to determine what kind of inputs you care about in practice.
Still heuristics can perform amazingly well in practice as we see with SAT or Integer Programming or Machine Learning. (PAC learning using the very simple model of 3-DNFs is intractable if NP$\neq$RP, and lots of experts think that RP=P). 915