An elf receives the benefits of a long rest in 4 hours while using the "Trance" trait.
According to the 2017 update to the Sage Advice Compendium:
Q: Does the Trance trait allow an elf to finish a long rest in 4
hours?
A: If an elf meditates during a long rest (as described in the
Trance trait), the elf finishes the rest after only 4 hours. A
meditating elf otherwise follows all the rules for a long rest; only
the duration is changed. This answer has been altered as a result of
a tweak to the rules for a long rest, which appears in newer printings
of the Player’s Handbook.
This ruling reverses guidance in the earlier version of the SAC, due to errata changing the rules for long rests.
Interactions between the "Trance" trait and long rests
A long rest is defined as:
... a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch.
The elf's Trance trait is defined as:
Elves don’t need to sleep. Instead, they meditate deeply, remaining semiconscious, for 4 hours a day. (The Common word for such meditation is “trance.”) While meditating, you can dream after a fashion; such dreams are actually mental exercises that have become reflexive through years of practice. After resting in this way, you gain the same benefit that a human does from 8 hours of sleep.
Since the "Trance" trait replaces the need for sleep (which most races need in order to complete a long rest), the elf is able to satisfy the requirements of the long rest while in a semiconscious trance for four hours.
It is unclear, however I lean towards it meaning that you can use it once between long rests.
"Once per day" occurs three times in the PHB, two of which have since been errata'd to mean you can re-use it after a long rest (and must have a long rest in order to re-use it):
Drow Magic (p. 24). Here “once per day” means you must finish a long rest to cast the spell again with the trait.
Infernal Legacy (p. 43). Here “once per day” means you must finish a long rest to cast the spell again with the trait.
This leaves Arcane Recovery as the only power in the PHB to refer to "once per day" without referring to a long rest. This may be intentional, but while "once per day" is used many times in previous editions (e.g. in D&D 3.5), there has been an effort to replace it by meaning to "once per long rest" in equivalent features. This works nicely for consistency, plus in most circumstances, it works similarly, as you can take only one long rest in a 24-hour period.
Additionally, on page 11 of the Monster Manual (under "Limited Usage"), it specifies that for monster abilities, if it says "1/day", the monster must finish a long rest to use it again.
So while this may not be the rule as written, I think it is one of those things that should have been corrected to mean "once per long rest", but hasn't (yet).
Best Answer
Short Rests are faster
There really is only one benefit of the Arcane Recovery feature over taking a Long Rest (4 Hour trance for Elves) and that is time. A short rest as defined by the PHB is:
If we then look at Arcane Recovery and what it grants us:
So the benefits of Arcane Recovery are the same for an Elf wizard as they are for any other race of wizard. The benefit of the trance is allowing a long rest in 4 hours, it does not modify the utility of a short rest which only takes 1 hour.
Limitations
As pointed out by @Joel Harmon and Dan B. The rules on Long Rests apply to the Elf's trace feature and contain the following:
If you have 4 hours to rest and have used neither your long rest or Arcane Recovery within the last 24 hours then yes, the Arcane Recovery feature won't help you that much since you can regain all of your slots with a trance instead of just some with a short rest.
If you don't have 4 hours, have already had a long rest, or can't convince the rest of your party to wait that long then Arcane Recovery will help you just like it will help any other wizard.
Since both spell recovery methods are limited to once per 24 hours it makes sense to use both of them to regain the maximum number of spell slots you can.