I want a Wi-Fi dongle that, besides acting as an ordinary Wi-Fi client, could change its mode to work as an Access Point too. Should I get a dongle with SoftAP or should I get one with Wi-Fi Direct? What is the difference (if any) between these two concepts?
Any Wi-Fi client can join an AP (including soft APs), but Wi-Fi Direct is a new thing as of roughly this year, so there are lots of devices that don't support it yet. So if compatibility with the largest array of client devices is a goal, go with SoftAP.
Wi-Fi Direct is a way for Wi-Fi devices to talk to each other directly, without the need to get them both associated to the same AP (or multi-AP network). If all the devices you cared about supported Wi-Fi Direct, then you could probably use it in lieu of SoftAP.
One additional advantage to Wi-Fi Direct in some scenarios is that you can use it even while each of your devices are connected to different wireless networks. Let's say you're at a conference chatting with a colleague, and each of you has his laptop associated to his own smartphone's personal hotspot or MiFi. Now you want to send a large file to your colleague (too big for email), and neither of you wants to lose his Internet connection during the file transfer. With Wi-Fi Direct, your two machines could establish direct Wi-Fi communication with each other without losing connection to the personal hotspots.