The items are similar, but there is a difference

Both of them allow spontaneous casters to get another spell to cast. Since they tend to have fairly hard limits on the spells known, this can be a great boon as there are many great low level buffing spells.

The Ring gives access to spells of other arcane casting classes, the Page doesn't

The real difference comes down to this from page of spell knowledge

If the bearer is a spontaneous spellcaster and has that spell on her class spell list, she may use her spell slots to cast that spell as if it were one of her spells known.

Meaning that it can only grant class spells. The Ring of spell knowledge does not have this limitation:

Arcane spells that do not appear on the wearer’s class list are treated as one level higher for all purposes (storage and casting).

The Page may hold a divine spell, the Ring cannot

Also, the page can be either arcane or divine, while the Ring is only arcane.

You can change the spell inside the Ring, but not on the Page

So why a Ring when it limits you to only arcane spells? Because you are able to change what spell is loaded into it (supported by this answer). So keep a scroll and if needed copy its spell into the Ring and there you go.

Arcanists are prepared casters

Though their unique spellcasting method is similar to other spontaneous casters, like the sorcerer or oracle, the only mention about spontaneous casting is this:

Like a sorcerer, an arcanist can choose to apply any metamagic feats she knows to a prepared spell as she casts it, with the same increase in casting time (see Spontaneous Casting and Metamagic Feats).

Using the definition on the core rulebook (magic chapter, page 206):

Most spellcasters prepare spells in advance—whether from a spellbook or through prayers—while some cast spells spontaneously without preparation.

Since arcanists prepare their spells, for all purposes and rules, they are considered prepared casters. And this is evidenced again on the Expanded Preparation feat, released on the same book.

Benefit: You gain an extra spell prepared of the highest level you can currently cast as an arcanist when selecting this feat. This is in addition to the number of spells you can normally prepare from your spellbook. You can instead add two spells prepared, but both of these spells must be at least 1 level lower than the highest-level spell you can currently cast as an arcanist when selecting this feat. You must choose which benefit you gain when you take this feat, and the extra spells prepared do not change level when you gain access to higher-level spells.

But if you are looking for official answers, there is an open FAQ request thread (with 40 votes) on their messageboards.