Finally, the match I've been waiting for! I think it is time to compare the two Dutch giants, and I'll start with carnivores.
Cats
Blijdorp of course has its rich history with cats of all sizes, but small cats in particular. Few species remain (serval, fishing cat and pallas cat), but those that do have nice enclosures, and the one for pallas cats is the most remarkable to me. Rotterdam has a strong collection of big cats, with Asiatic lions, Sumatran tiger and Amur leopard. None of these enclosures is a stand out in my opinion, and Blijdorp seems to have a thing of ugly fencing. It takes a second look to discover the charms that each enclosure has. Especially the lion exhibit is nice if you forget about the fencing.
Pallas cat, lions, tigers
Burgers' has only one small cat (bobcat), and the enclosure is decent but completely indoors. I does however, fit nicely within the Desert area. The big cat collection surpasses Rotterdam's, as Burgers not only has lions, tigers and (Sri Lankan) leopards, but also cheetahs. The lion and cheetah enclosure are great for the animals but look rather bare in winter, but it's a different story in summer. The tiger and the leopard enclosure are excellent. The latter especially, as it provides shelter, differences in hight and lush vegetation, and still offers good views.
Bobcat, cheetah, leopard
Canids
Blijdorp obviously wins on canid collection. Even now the wolf pack has died out, they still have arctic fox, maned wolf and swift foxes, the latter being the rarest carnivores of the park. All enclosures are good, although viewing is rather poor for the maned wolves. The arctic foxes have a great enclosure though. Unfortunately, both the maned wolf and the swift fox are due to leave the collection, leaving only arctic fox.
Arctic fox
Burgers has only one exhibit, housing breeding group of golden jackals, but the quality is very high. Build in roughly the same style as the leopard enclosure, it shares many of its qualities.
Golden jackal (no good enclosure pictures in gallery, unfortunately)
Mongooses
None of the zoos has an interesting set up for mongooses. Blijdorp though has two species, exhibiting not only meerkats but also yellow mongooses. Both have decent enclosures. Extra point for Blijdorp as the meerkat's indoor exhibit is visible.
Yellow mongoose
Burgers' has only meerkats, but the exhibit is arguably better than Blijdorp's. It has more live vegetation where the animals frequently forage. Another nice feature is the presence of a heat lamp in a fake tree, which attracts the meerkats to a spot close to the public.
Meerkats
Bears
Blijdorp's one and only bear exhibit houses polar bears. Although it has various shortcomings (as explained by
@lintworm), it is still one of the best enclosures for the species I've seen. Providing natural substrate and plants is a nice feature, and the huge viewing window can result in spectacular views.
Polar bears
Burgers', however, is also not weak when it comes to bears. Their sun bear exhibit is impressive, and offers multiple climbing opportunities, hight differences and lush vegetation. The views are great, and if it gets too cold, the bears have a visible indoor exhibit. Recently, even more climbing structures have been added.
Sun bears
Pinnipeds
When it comes to enclosures, Blijdorp dominates in this category. Their California sea lions live in an impressive exhibit, allowing for both views on land and under water. The pool is large and deep. Arguably one of the best carnivore exhibits in Rotterdam.
Sea lions
Burger's can only counter that with the species on display, as the pool housing ringed seals (which by the way, is much deeper than one would expect) is one of, if not the worst carnivore exhibit in the zoo. The very rare ringed seals however, do breed there.
Ringed seals
Procyonids
Blijdorp has one species of procyonid, the common raccoon. The enclosure however is nicely themed and gives the animals both opportunies to climb as well as shallow water area and vegetation.
Northern raccoon
Burgers' coati exhibit rivals Rotterdam's raccoons. It is huge (in fact a former tiger enclosure), well planted, allows for digging and with lots of climbing frames. But, as we all know, Burgers' has another procyonid, the rare ringtail. It is exhibited in two nice enclosures in the Desert. Especially the larger of the two allows for a full range of natural behaviours.
Ring-tailed coatis
Other carnivores
Blijdorp has representatives of two families that are absent in Burgers'. Spotted hyena is one of them, but as their enclosure is being/has been renovated I can't commend on exhibit quality. The new red panda enclosure is very good though, if only those pandas just stayed inside it
Red panda
Burgers' has three families that Rotterdam doesn't exhibit. The striped skunk lives in the same enclosure as the ringtails. Then there are the Asian small-clawed otters in the Bush, housed in a large enclosure, and the binturongs living alongside the sun bears. The otters have the excellent enrichment opportunity of finding a breeding fruit dove in their next box
Not so excellent for the dove naturally.
Otters
Thanks for the extensive photographic series by
@vogelcommando and
@Toddy in the gallery, which capture the far majority of the exhibits.
16 species each it is. 
! ).
