Wow, you don't want much there... This answer comes from a PhD chemist whose specialty was inorganic analysis using spectroscopy and computer real-time data capture. Hold on to your seats for this one.
The biggest difference stems from a combination of atom size and the ability to form complex non-uniform compounds based on covalent bonds (see below for more) leading to long chains of atoms. If you look at the periodic table, the atoms most likely to engage in organic compounds are in the top two rows towards the right side of center. Here is a link to the Wikipedia article.
https://en.wikipedia.org/wiki/Periodic_table
From a chemist's viewpoint, it's all about electrons and "orbitals" - which are technically both hard and easy to understand depending on how deep you want to dive. There is a property called electronegativity or electropositivity (related to electron affinity) that relates to how strongly a particular atom donates or accepts electrons. The greater the difference in electronegativity between component elements, the stronger and more intense the bond.
Basic factor: Orbitals are more stable when they are complete or empty. In between? Not so good. One orbital holds two electrons due to quantization. Each "layer" in the periodic table leads to atoms having more orbitals, and the capacity for more orbitals depends on physical size of the atom. The farther down you go on the periodic table, the physically bigger the atoms become, and eventually that size precludes the chance for the atoms to form convalent bonds. They can't get close enough. (Note: I am trying to "dumb this down" so as to not lose everyone instantly.)
Acid/base reactions can be defined two ways. The Bronsted definition is "hydrogen atom donor/acceptor" but the Lewis definition considers the ability to donate or accept electron pairs as part of the compound. Obviously, the Lewis definition is tied into orbitals. When a compound acts like a Bronstead acid or base, it is more likely to form ionic compounds.
Ionic bonds form between things from opposite sides of the periodic table. Salt is sodium chloride - one element far right, the other far left - and they form a very strong bond referred to as IONIC. (No relation to Greek column style.) Ionic compounds form rigid crystals. They don't flex - they shatter under stress. Sodium can lose an electron OK because in so doing, it loses an incomplete orbital. (Why it is "incomplete" starts to get complex.) Chlorine can happily gain an electron OK because in so doing it forms a completed orbital. (Again, complex.)
Covalent bonds form between things closer together in the periodic table. Carbon, Nitrogen, Oxygen, and to a lesser degree, Sulfur are all able to share electrons. Silicon also has this ability but its size begins getting in the way. This sharing is based on each partner of the covalent bond having joint custody of enough electrons to form complete orbitals even if they don't have sole custody of the electrons. When this happens, if you want to try to build the mental picture, the "electron cloud" around the participant atoms is such that the shared electrons have a non-trivial probability of being associated with either of the participants. This is the minimal form of what we sometimes called "delocalization" - though this isn't the strongest form of it. Think of the electron as a kid bouncing from dad's house to mom's house, but in this case everyone is happy about it.
Try to visualize this next part. Consider benzene, which is C6H6. August Kekule determined the structure of benzene, which is a flat hexagonal ring of Carbon atoms, with one Hydrogen atom sticking out of each Carbon and two bonds between the two adjacent Carbons in that ring. But when you do the math, there is another electron per Carbon atom and it appears to be totally shared among ALL SIX carbon atoms. The effect that has is that the electron's probability of being on any particular Carbon atom is 1/6th. It is said to be totally delocalized. Since every carbon is sharing one electron that way, there is a high probability that the Carbon atoms have three full orbitals and three empty orbitals, both of which are stable situations.
Take this image up to graphite, which is essentially a flat plate of Carbon atoms interlinked like a honeycomb. No hydrogen involved. Every Carbon has three bonds to three adjacent Carbon atoms. Three bonds - but Carbon has a fourth electron to contribute. Graphite has electron "clouds" of incredibly delocalized electrons and each sheet of hexagonally arranged Carbon atoms "floats" on the electronic repulsion of the electron clouds above and below each sheet. Sort of like the old "air hockey" game where the plastic puck floats on air and doesn't really touch the flat surface of the table. And it is totally covalent.
If you have ever heard of graphene, it is related to graphite; engineers have found ways to take advantage of that delocalization as a way to conduct electricity with low resistance. Carbon nano-fibers are another example of highly delocalized bonds to give strength to the compound. Also look up "Bucky balls" which are incredibly interesting Carbon compounds.
Back to the question of the difference? Versatility. Organic molecules contain atoms that are small enough to get up-close-and-personal with each other. Inorganics, due to either size or "unwillingness to share electrons," lack the ability to form intimately small compounds.