Yes, the 2.13GHz MacBook Air will smoothly run flash apps and 720p avi files. And I agree that the size and weight is attractive. But I would recommend the 13", 2.53GHz MBP. Though I guess it really depends on your needs and what you expect from a notebook. My main reasons for recommending the MacBook Pro are usability and performance.
The 13" MBP comes standard with 4GB of 1066MHz DDR3 SDRAM; upgradable to 8GB. The Air maxes out at 2GB. The MBP also has the new 7hour battery, the Air will on get 5hours. The MBP also has a faster, more powerful processor. You also have the ability to use larger and faster HDDs or larger SSD in the MBP whereas you're limited to a 128GB SSD with the 2.13GHz Air. Moreover, the MBP has two USB ports, a FireWire 800 port, SD card slot, Gigabit Ethernet port, Mini DisplayPort, and a SuperDrive (DVD±R DL/DVD±RW/CD±RW); the Air only has one USB and a Mini DisplayPort, no internal optical drive. Also, the MBP is a few hundred dollars less than the 2.13GHz Air.
I'm sure the Air will support smooth video playback while you have a (modest) number of applications open. But the MBP is a much more powerful machine. I have a 15" MBP with 4GB of RAM and also run countless applications simultaneously. I've yet to notice any lagging. The safe bet, in my opinion is to go with the MacBook Pro. It's cheaper, it's more powerful, it has an internal optical drive, and it's nearly as sleek and portable as the Air--if you're near an apple store, you should take a look at one in person, it's lighter and more portable than one imagines it would be.
Also, you can purchase the base model of the MBP now and in a year or two, invest a little more money to upgrade the RAM, perhaps install a larger HDD/SSD as newer models are released. What I mean to say is, I feel the MBP will have a longer life; you won't feel as if you have an archaic obsolete machine in a few years.
I do love the sleekness and portability of the Air, but for me, the performance just isn't there. With the addition of the 13" MBP to the Pro line, I feel Apple has really bridged the performance-portability gap.
My best advice would be to take a trip to an apple store, or any computer retailer with both a MacBook Pro and an Air on Display. Take them for a test drive. And see which works best for you. Good luck and enjoy the new computer, whichever you finally decide upon.