This is tough for me to answer objectively, but I'll do my best.
Both services offer many of the same features and eliminate the pain of managing your own email server. Both services will achieve better deliverability at scale than managing your own server unless you know what you are doing and most people aren't interested in diving that deep into email.
Mailgun is very focused on serving developers and that guides our product roadmap. Therefore, we don't offer features like WYSYWIG newsletter creation tools. We build everything API first and we strive to make them RESTful, intuitive and well documented. Admittedly, our GUI / control panel is not the strongest part of the product. We also focus a lot on incoming email and we are pretty proud of our inbound email parsing through Routes. Mailgun is owned by Rackspace (acquired in Aug 2012).
Sendgrid is definitely the largest transactional email sender in the market. I recently read they do something like 7 billion emails monthly. They have more tools for non-developers like newsletter creators and support language wrappers that some developers prefer. They also offer inbound through their Parse App. Sendgrid is an independent, private company that has raised $27.4mm in funding according to Crunchbase [1].
My recommendation would be to try both - the docs are all online and both offer free plans to test with.
Mailgun sounds like a really good fit for me. I'm running a couchdb server, and I want to set it up to POST to my DB. Dump straight into my document store, life is good.
I did a quick try at it and ran into a 415 from couchdb, which means mailgun didn't send json. Is there a way to configure a route to do that? If not, I'll have to write a custom _update handler.
Our SendGrid experience has been middling. We have fought a half dozen of bugs -- the week I was implementing it our first app I had seven tickets open in their support system -- with admin screens that time out or don't work, APIs that time out, etc. Everything from the APIs to the admin just feels a bit unfinished and amateurish.
Support has been very lacking. At one point I had waited a long time (a week?) for a ticket response while they said they would investigate, and suddenly they just closed the ticket with no explanation. I felt like I had to fight to get any attention.
That said, its delivery seems reliable.
We have since switched to Mailgun for a lot of stuff, and will be migrating the rest soon. Mailgun has been rock solid from the start, the admin screens are fast, the APIs just make sense, and they were really responsive when we asked about a missing feature. Unlike SendGrid where the delivery log took ages to show results, Mailgun's equivalent page is really snappy. A big time saver when someone complains about not receiving a password or something.
