The exact field of view (FOV) of a telescope is often not a unique number, but depends on the instrument in use. For JWST, the FOV is a typically few arcmin (1/60 of a degree). Here you can see the possible FOVs for all JWST's four instruments, NIRSpec, NIRCam, MIRI, and NIRISS/FGS:
Fields of view of JWST's instruments, measured in arcsec, i.e. 1/3600 of a degree. Credit: JWST User Documentation.
As you say, compared to JWST (indeed, to most other telescopes), Euclid has an enormous FOV. Its two instruments, VIS and NISP, have nearly the same FOVs of 0.75×0.75 square degrees, or 45×45 arcmin2.
A few weeks ago, the first scientific studies from Euclid were released. One of my favorites is this image of the galaxy cluster Abell 2390 and its surroundings. The zoomed-in part shows roughly the FOV that JWST would see:
Credit: ESA/Euclid/Euclid Consortium/NASA, image processing by J.-C. Cuillandre (CEA Paris-Saclay), G. Anselmi; CC BY-SA 3.0 IGO or ESA Standard Licence.
Why are they different?
The reason for the wildly different FOVs is that the two telescopes have very different science goals. JWST's main purpose was, originally, to look for the very first galaxies. As these galaxies are expected to be small, but numerous, are large FOV is not needed. What is needed is, in addition to being sensitive to infrared because the light is so redshifted, to catch as many photons as possible, since these galaxies are so extremely faint, being so far away. That is, a huge mirror. The galaxies are faint, but numerous, so one can be certain to have many of them in the FOV, no matter which direction we look.
Another goal, which was added later, is to probe (the atmospheres of) exoplanets, which also doesn't entail a large FOV.
On the other hand, Euclid is built to observe a huge number of galaxies throughout most of the history of the Universe — but not its earliest epochs — in order to probe how the Universe has expanded through time. Another goal is to observe a statistical sample of the most massive galaxies in the Universe. Since galaxies are rarer, the more massive they are, one needs to observe a large area on the sky to get them. In contrast, JWST is unlikely to catch even a single one in one pointing.
Why not both?
So why not just built a Webbclid Telescope that peers as deep as JWST over the area of Euclid?
In astronomy, even bright objects are usually faint enough that you want as many photons as possible. You get this by 1) exposing for a long time, and 2) using as big a mirror as possible. The larger FOV you spread these photons over, the fainter your image will be, and hence a large FOV means that you need 1) to expose for longer, or 2) have a larger mirror.
JWST's FOV is $\sim20^2$ times smaller than Euclids, and its mirror is 6.5 m in diameter. For the same exposure time, Webbclid would therefore need a mirror with a diameter of $\sim 20\times6.5\,\mathrm{m} = 130\,\mathrm{m}$.
That is
what prevents us from building a telescope with the same resolution of JWST that can see as far in the past as JWST but with the field of view of Euclid.
As an analog, when you use a projector to project your movie on the wall, you can get a larger screen by moving the projector farther from the wall. But this comes at the expense of getting a fainter image.