We use both on a regular basis. We have use the tripp trapp at home and an Antilop chair stays at the grandparents for when we visit.
The tripp trapp is so comfortable for our 9mo, and our 3yo also has one - he find it so much easier to eat at the table when he's at the right height (obvious, but true).
I really can't think of any disadvantages with the tripp trapp - it's got a small footprint and can be pushed right under the table when not in use. The baby set cushions are a PITA to get on and off when they need cleaning. I've just read upthread about just keeping the back cushion on - good idea - I may just do that.
Antilop: great chair but its legs splay out - I'm forever tripping over them when we use it. Very easy to clean though, and takes apart in seconds. I do find there is a huge gap between the inner edge of the tray and 9mo's tummy - most of her food ends up on her knees! You don't have to use the tray though, and it will push up to a table, but it won't necessarily be at the right height.
Tripp Trapp do their own food tray which I think others upthread have referred to here.
The playtray which, I think OP you referred to here isn't actually made by Tripp Trapp, but as the whole Stokke 'ethos' is to use the table, they don't actually do their own. This Playtray was specially designed to fit the Tripp Trapp, though. We bought one when our 3yo was a baby. While it is a bit expensive, we really couldn't have done with out it and our 9mo uses it now. The advantage is that they can sit playing with toys or snacking in the highchair without having to be at the table, and it's easy to clean and can be thrown in the dishwasher. A bit of an investment for a bit of plastic, but well worth it IMHO!
On balance, if money isn't an issue, then go with the Tripp Trapp and a tray. If you're short on space, go with the Tripp Trapp. If you're short on space and money and don't mind dismantling now and then, go with the Antilop.
Sorry, bit of an essay, but HTH! 