The AIM mission will actually be an exciting one. There's a binary asteroid heading broadly towards us (closest pass in 2023 at 0.1 AU or so). The focus of the mission is on the smaller of the two asteroids.
ESA will send an orbiter towards it around 2020 with a mission profile broadly similar to what Rosetta is doing at 67P/C-G. That is, examine it from various orbits, find some specialties, put a lander on it (a MASCOT derivative - like the one currently enroute to another asteroid with Hayabusa 2). Also check out the big asteroid as well. As a bonus, they're bringing along a number of cubesats as separate orbiters for which they're currently holding a competition. From the ESA side, this mission satisfies two goals: one, we're doing our own mission to an asteroid, and two, we're doing some technology testing - those cubesats would be the first on any such mission, and the orbiter will test a laser communication terminal at interplanetary distances. The mission is moderately small - about 260 million Euro including launch, but without the instruments that are currently only considered as a black box strawman package. Those will involve radar and IR measuring, broadly, some in conjunction with the lander.
The NASA mission (DART) has a different focus: Its sole intention is to produce an impact of about 1.33 tons TNT-eq (300 kg at 6.1 km/s) on the surface of that smaller asteroid. The impact will change the orbit of the 150m-diameter asteroid (around its binary partner) slightly in such a way as to be measurable even from Earth. The only real instrument brought along is a flight spare of the LORRI imager from New Horizons used for navigation and some images before impact, hence virtually free. The launcher is not included in NASA's extremely low 150-million USD estimate for the mission, but given that it'll likely launch on a ULA rocket, will be somewhere around twice the mission budget (the larger Deltas run up to 450 million per launch).
Both missions are built in such a way that they can function completely independently - basically the ESA orbiter will just be conveniently in place (withdrawing to a safe spot 100 km away) to observe the impact, measure its results and reexamine the asteroid for impact damage on and under the surface. NASA can pitch it as a cheap mission to test technology to deflect asteroids (at least that's how they're pitching it), ESA can pitch it as "see, this is the same asteroid impact study mission we already planned five years ago - but now for less than half the money".