The Heer (the army part of the Wehrmacht) was a conscripted force, very well trained, especially early war and generally the regular force of the nation of Germany.

The Waffen SS (the armed, fighting part of the SS) started out as a volunteer force. Some would argue that the nazis created and expanded on the SS since the army resisted attempts to tie it closer to the party rather than the state.

Initially, the SS was treated very stepmotherly by the army. They were not allowed to purchase arms from the same channels as the army, and were forced to use hand-me-downs such as Czechoslovak rifles and light and heavy machine guns, or the horrible Knorr-Bremse MG 35/36. Instead of MP 38 SMGs, they had to use ex-Austrian MP 30(ö) or MP 34(ö) or even close to useless Mauser Schnellfeuers. Initially, they were not allowed tanks or artillery. An SS artillery regiment with either hand-me-down 10,5 leFH16 or modern 10,5 leFH18 (I have been unable to confirm which of the pieces they had initially).

During the 1939 campaign in Poland, the SS was absolutely not an elite force. The SS Germania regiment ran in the face of a counter-attack by a Polish regular infantry regiment, abandoning all heavy arms. The SS tried to replace tactical skill and flexibility with zeal and discipline with very bad results.

Before the 1940 campaign in the west, the SS had hired army instructors to train them in infantry tactics and had formed several elite motorised regimental groups that were attached to Heer divisions. They also had a division of their own, which was equipped with Czechoslovak artillery and had a mediocre performance against the Dutch.

From now on starts the era when the SS was elite. It is short - the SS partook in the invasion of Yugoslavia and Greece, again as elite motorised regimental groups and now mostly equipped with the same arms as the Heer. An all volunteer force, highly motivated and trained, they can be called an elite force from early 1941 to mid-1942. That is when the SS started to form its own divisions.

The first 5-6 SS Divisions were quite good, another 3-4 decent while the remaining 20-30 (some were divisions in name only) were quite bad. From late 1942 onwards, the SS raised multiple very bad formations capable of little more than atrocities and anti-partisan fighting.

SS units were never that much better than the best of the Heer units and average quality would only be better for a short while (1940-42). Later in the war, "volunteers" were dragged from POW camps, conscripted from occupied countries (such as Estonia and Latvia) or forced to volunteerat gunpoint.

The Western Allies only faced the best SS units, and often when they were resting or sent in full-strength (or almost full-strength) to help a situation, and thus only experienced the best of the SS. This contributes a lot to the myth of the SS as an elite force.

There were two main divisions within the SS, only one of which is strictly relevant to your question:

• the Allgemeine-SS ('General SS'), comprising Himmler's staff, the bureaucratic/administrative departments like RuSHA (the office responsible for racial purity), and the security forces, including the Gestapo, SD and concentration camp guards. This is the SS division which administered the camps, enforced Nazi ideology and functioned essentially as a state-within-a-state.

• the Waffen-SS ('Armed SS), which was a military formation organisationally independent of the Wehrmacht. The Waffen-SS eventually grew to number dozens of divisions.

As /u/Xfbb said below, the Waffen-SS divisions weren't initially intended as front-line combat troops, but were quickly placed under OKW tactical command at the outbreak of war. SS units operated at the regimental/divisional level within larger Wehrmacht formations.

The Waffen-SS certainly regarded themselves as an elite force, above the regular Heer troops. But they were funded and equipped (in large part) by the OKW, so there are limited differences in equipment or military doctrine — it may be that SS units, as 'elite' forces, were better equipped, but I don't have any sources for that.

Other notable differences, in broad answer to your question: the SS maintained a separate rank structure to the Wehrmacht, and recruited separately — if you look at the essay I linked earlier, there's some interesting detail about the SS' stringent recruitment process, which stemmed from their original role as Hitler's personal guard.

The SS were significantly more ideologically driven than the regular Wehrmacht; they were, after all, a party institution rather than a state one, under Himmler's direct authority and with the express purpose of protecting the Nazi state and its leadership, and enforcing its ideological goals — hence the SS' role as the prime mover in Nazi war crimes and genocide.

It was SS organisations — the RHSA — which were responsible for maintaining Nazi Germany's internal security and crushing opposition, and SS units which administered the concentration camps and the Final Solution. Those are strongly ideological roles, quite distinct from the generalised warfighting role of the Wehrmacht.