When it comes do overall combat effectiveness between these two species, there isn't a stark difference between the two such that one will always win against the other and that proper use of strategies tactics, logistics, availability of resources and others would be more significant factors to dictate how a battle would go.
However, when you compare the two groups individually, or one squad of Brutes vs one group of Elites, that's where combat doctrines, the species' culture and biology begin to factor in more, especially in certain situations. Take close quarters combat for example pitting an unarmoured Brute and Elite against each other. In this case, there's little doubt that the Brute would massacre several Elites in close quarters due to their superior size, strength and speed. We see in Halo Wars 2 that Brutes are strong enough to throw around Elites AND Spartans across the room with little difficulty. While the Elite may be trained and disciplined, the strength and size gap between the two is too large for training to close. Furthermore, we know that Sangheili combat harnesses augment the user's strength not unlike MJOLNIR, so a naked Elite is at a further disadvantage, and even with armour Halo Wars 2 shows us just how powerful the Jiralhanae are in close quarters.
Overall, a 1v1 wrestling match would favour the Brute significantly.
Next question is, what about a 1v1 fight with the Elite equipped with a plasma rifle and armour while the Brute is equipped with a spiker and Brute Power Armour? In this case, I'd probably give the Elite a slight edge given that given their training and discipline the Sangheili would probably be better marksmen and with regenerating energy shields the Elite could probably outlast the Brute given that Brute Shock plating falls apart when it's overloaded (see Halo 3 and Halo: Retribution). However, I wouldn't write off the Brute entirely. Brutes still possess the physical advantages it possesses in their greater size and running speed, and are equipped with heavy armour plating weighing at least half a ton. The Brute Spiker as well is an excellent weapon that can eviscerate the Sangheili. A single round is enough to pin a marine into a concrete wall and we see in Halo: Broken Circle Brutes use the Brute Spiker to great effect.
Overall, in a one vs one firefight, it depends more on the individual Brute/Elite's skillset and who can utilise them better. The Brute can withstand a great amount of punishment before going down, while the Elite's energy shields would help them last longer in a firefight as long as they are up and running.
So that leaves us in a squad vs squad battle. In this case, I think it's important to look at the cultural mindset of both species. Looking at the Sangheili, the Sangheili hold honour above all else, and the way they progress through the Covenant military is largely based on how many kills an Elite has. This in turn makes the Sangheili very individually driven and would thus helps makes each skilled and competent Sangheili stand out amongst his peers. However, that honour based mentality can also act as a crutch. Because the Elites are very individually driven, many individual Sangheili would be more interested in their personal advancement rather than doing something for the benefit for the group, and would act independently from the rest of their peers. In comparison, the Jiralhanae operate under a strict social hierarchy and rigid pack mentality, making them excellent pack hunters. Because of the Jiralhanae's pack mentality and social hierarchy, the Jiralhanae are better able to cooperate with each other and they are so effective as a team that UNSC pilots actually fear Jiralhane Seraph squadrons over Sangheili ones as mentioned in Halo: Warfleet because of these traits, and I'd imagine that the same can be applied to other areas of combat for the Jiralhanae. Fred-104 even notes in Halo: Retribution that the Jiralhanae have perfected a "rolling ambush" where a Jiralhanae ambushes a target from the rear and pushes the target into a killing field occupied by the rest of the pack. Furthermore, unlike the Sangheili, the Jiralhanae are pragmatic and utilitarian in nature and have no problem using whatever resources and tactics they could use to overcome an enemy. For example, Fred-104 notes in Ghosts of Onyx that an Elite would rather fight with an empty plasma rifle rather than pick up a fully loaded human assault rifle, while the Jiralhanae are more than happy to pick up a UNSC shotgun or scavenge UNSC arms and equipment and repurpose them for their own use, as seen with the Brutes in Halo 2, or the Banished, respectively.
Because of the Jiralhanae's social structure and mentality placing a huge emphasis on teamwork and cooperation between their peers as well as the Jiralhanae's pragmatic and utilitarian nature, I would put this fight in their favour.
Though of course, these are general cases and while one side may have one inherent advantage over the other doesn't mean that things always swing that way. We see in Halo: Tales from Slipspace - Hunting Party that a very capable team of Special Operations Elites - the Silent Shadow - can massacre dozens of Jiralhanae without too much issue, just as we could witness a single Brute smack around a bunch of Spartans like a bunch of toddlers.
However, if I had to say which one is better than the other, I'd personally lean more towards the Jiralhanae as, when considering an equally skilled Brute and Elite against each other, the Brute has the biological and even some cultural advantages over the Elite and that when you begin looking at peak examples of each species, an Elite would just be comparable to a Spartan (e.g. the Arbiter, or Rtas 'Vadum) while a peak Jiralhanae can overpower several of them (e.g. Atriox).
As to sources, I suggest viewing these two compilations of feats on the Jiralhane