Two different jobs
Both are home practice aids for putting, but they train different things. A putting mat recreates the experience of putting - a surface to stroke a ball across, a hole or automatic return at the end - so the golfer grooves a repeatable stroke and gets a feel for distance. A pressure putt trainer is a focused device: a small ramp that returns the ball to you only if you strike it with perfect pace and line, and sends a mishit away. One rehearses the whole motion; the other drills precision.
The putting mat: realistic reps, if there's room
A mat is the closer thing to actual putting, which makes it the better pick for a golfer who wants to hole out over a realistic distance and build a consistent stroke. The catch is space - a good mat wants a hallway or a clear run across a room, and it has to be somewhere it can stay rolled out to actually get used. For someone with the room, it's the more engaging of the two and the one that gets picked up most often.
The pressure trainer: precision in no space
A pressure putt trainer earns its place on feedback and footprint. Because it only returns a perfectly-struck putt, it's brutally honest about pace and start line, and it fits in a few feet of an office or living room. It won't teach distance control over a long putt, but for sharpening the short ones and getting instant feedback in a tiny space, it's hard to beat - and it's the easier one to leave set up permanently.
The honest caveat for both
Neither works if the recipient won't practice. A training aid is a wonderful gift for a golfer who genuinely wants to improve and a dust-collector for one who plays purely for the social round. So the real question isn't mat versus trainer - it's whether this person is the type to putt on the carpet on a wet evening. If they are, either lands well; if they aren't, a consumable is the better gift.
Which to buy as a gift
Buy the mat for a golfer with floor space who wants realistic, engaging practice; buy the pressure trainer for someone tight on space or focused on sharpening precision, and for the gift that's easiest to keep set up. Both are affordable enough to pair with a small extra - a sleeve of balls, a marker - to round out the gift. For the full range of practice gifts, see best golf training aids.