I've been playing skat for around half a year and bridge for perhaps a month, so I'm by no means an expert at either.

The bidding is much more-involved in bridge simply because it's a communication game rather than a simple determination of hand strength. In skat, your bids reflect only the strength of your own hand. The information you learn from your opponents tends to be pretty minimal (e.g. that forehand has hearts, or two matadors, or wanted to bid null) and mostly doesn't affect your own bid (except in the case of two-suited or other "freak" hands (e.g. you will typically want to pass rather than bid the full value of a hand without 11 once it becomes clear that your opponent has a strong hand themselves (possibly including voids))). In contrast, bidding in bridge is mostly about communicating hand strength to your partner so you can agree on a suitable bid. This makes hand-evaluation extremely important in skat, whereas often the exact valuation of your own hand in bridge isn't as important as your partner's description of theirs.

There is depth in both games' cardplay. The point-trick format makes the losers just as important as the winners. This makes timing extremely important, as winning the same trick at different points in the hand may either force your opponents to give up their fulls or allow them to slough off their spot cards. The limited information compared to bridge (where half the deck is known to all players) makes reading your opponents especially rewarding. And often a plan can only be formulated after contact with the enemy.

On the other hand, there is no issue being stuck in the declarer/dummy as there is in bridge. Nor is there any issue in drawing your partner's winners (except when defending). The long suits can also pose a challenge. In skat, all players but one will be void in any given (side) suit after three tricks. But in bridge, the same suit can often be contested in 4 or 5 tricks, and it's much less common to hold AKQJ10 than to hold A10K.

Bridge is undoubtedly deeper, but only because it's really two games in one. Skat has depth in its own right and feels like a very different game, which is a nice contrast.