We're a Tanzanian company, so read this knowing that, and we'll keep it straight: both countries are excellent, and the right answer depends entirely on your month.
One ecosystem, one border
The Serengeti and Kenya's Masai Mara are the same ecosystem with a national border across it. The animals ignore the border. The Serengeti is much the larger part of it, and the Mara is the smaller northern extension.
Where the herds actually are
In Tanzania for most of the year. The calving plains, the western corridor, the Grumeti and the northern Serengeti are all Tanzanian. The herds push into the Mara in roughly August and September and head back south from around October. So the Mara gets the herds for part of two or three months; the Serengeti has them for the rest of the year.
So why is Kenya so associated with it?
Because the Mara River crossings into the Mara Triangle are highly concentrated, very accessible, and that's where a great many of the photographs and documentaries are made. A small reserve with a lot of animals and a lot of camps produces a lot of pictures.
The practical differences
Distance. The Mara is a short flight or a long drive from Nairobi. The Serengeti is bigger and spread out, so you'll fly between areas or accept long drives.
Crowds. The Mara in peak season is dense with vehicles around sightings. The northern Serengeti in the same weeks is busy by Serengeti standards and quiet by Mara ones.
Rules. Off-road driving is permitted in parts of the Mara conservancies and not in the Serengeti National Park, which matters a lot to photographers and not at all to most people.
Cost. They're broadly comparable. Kenya has more budget options near the Mara; Tanzania bundles better with Ngorongoro, Kilimanjaro and Zanzibar.
How we'd choose
Travelling in August or September and the crossings are the whole point: either works, and the northern Serengeti gives you the same river with fewer vehicles. Any other month: Tanzania, by a distance. Want the migration plus a crater, a mountain and an island: Tanzania. Short trip from Nairobi: Kenya.
Doing both
Possible, and it costs you a day and a border. It's worth it on a long trip and it's a waste on a ten-day one, where the extra days would do more in one country.
The conservancies, since people ask
Kenya's Mara has private conservancies on its edges where vehicle numbers are capped, off-road driving and night drives are allowed, and the land is leased from Maasai landowners. They're excellent, and they're the real answer to the Mara's crowding problem. Tanzania's equivalents are the concessions and game-controlled areas bordering the Serengeti, which work the same way and are less well known.