How much storage does the Collector's pass add?

That is the top end of what the companion passes offer. The Advanced tier gives five extra boxes, so if storage is the thing pushing you toward a paid tier, the Collector's tier is the only one that goes further. You may also see the same perk written as 10 storage slots rather than ten boxes. It is the one grant, described two ways.

Storage matters because catching is the loop. Boxes are where everything you are not carrying with you lives. Buying for the boxes alone is a judgement call, since the storage arrives as part of a $20 bundle and is not sold on its own.

What else comes in the $20 tier

The Collector's Edition pre-order sits at $19.99 to $20 depending on where you buy it. Alongside the boxes, activating it hands you an instant 10-level boost, so the account you start on is not a fresh one. There is also a monthly pass component that pays out currency across a 30-day period rather than all at once. The exact figure moved around during the beta and has appeared as 3,000 Glimmer, as 300 premium currency, and as 330 Lumen crystals, so treat the drip as settled and the number as not.

Not settled: Per-box capacity has not been stated, so ten boxes does not yet convert into a known number of Aniimo you can keep.