What is the difference between a cafe and a diner?
Basically, a café is (in the US) a place you go for coffee and other beverages and maybe a light meal, such as a sweet roll or perhaps some pie. However, many cafés will also offer a sort of lunch menu with sandwiches and possibly burgers, fries, etc.
A diner is a place with offers full meals, generally of a relative proletarian nature (versus what would be served in a high-falutin' restaurant). Typically burgers, fries, modest steaks, maybe some fish, a limited selection of salads, and, of course, an assortment of beverages, centered around coffee and soft drinks.
The diner would typically have waiting staff, while the café might be either a server or (stand up) counter service. But either might (or might not) have a sit-down counter, in addition to tables. Typically the café is open for breakfast and lunch only, while the diner is open for lunch and dinner, and maybe open for breakfast, maybe not.
A fast food place would have fare similar to a limited diner, only no waitstaff.
(In fairness--not that New Jersey deserves it--there is a lot of variability across the US as to what these terms mean. My description above is probably most fitting to rural establishments, while metropolitan areas will develop their own quirks as to meaning.)
To make it as simple as possible: they are quite similar, but with a different focus.
The connotations are going to vary from region to region and person to person, so adding a great deal of detail is kind of pointless. Here's the general gist, which would be accurate everywhere I've ever been in the US:
You go to a cafe for coffee, and might get some food while you're there.
You go to a diner to eat a meal (dine), which will probably involve a beverage, which could be coffee.
In the US, a diner has three characteristics features: a window counter through which you see the cook/kitchen, booths and a counter with stools. That is the classic American diner. Funnily enough, in cities like New York, they are today almost all owned by Greeks or descendants of Greeks. They have traditional American breakfast, lunch and dinner foods. Nothing too fancy. Caters to all income types and is typically not expensive. Traditionally, also diners were modelled on dining cars from trains. Here is a "typical" diner, in Minnesota: https://en.wikipedia.org/wiki/Mickey%27s_Diner The traditional ones (from the thirties) were often art deco.
A cafe is associated with the idea of little tables (sometimes placed outdoors) where you can partake of slightly fancier fare or "cuisine" (salads and omelettes and dishes cooked by a chef) and is more European (French, Italian etc) in feeling. Often, they also serve wine and beer. They are rather informal but the good is usually good. Layout and atmosphere is more sophisticated, usually, than a diner. You don't get truck drivers, for example, sipping white wine of a summer afternoon. Here is one in Washington, DC https://commons.wikimedia.org/wiki/File:HK_TST_1881_mall_restaurant_DG_Cafe_outdoor_sidewalk_umbrella.JPG
Food-wise, a cafe is more "sophisticated" than a diner. Diners are associated with typical American breakfasts, sandwiches for lunch and things like meatloaf for dinner. There is, of course, overlap.
I don't know why that Saskatoon Park Café and Diner has both words. Most likely, they want to appeal to both the cafe crowd and the diner crowd. They have pretty fancy sandwiches that go beyond typical diner fare....
http://www.parkcafe.ca/menu