HTTPS gets you confidentiality (encryption), authentication (identity), and integrity (tamper-evident connections).
You don't care about so much about the first one in your case, but you should care about the second two. The "identity" part implicitly protects you from certain DNS attacks, but there is the chicken-and-egg problem (hence HSTS). There's also the "green URL bar" and all that nonsense, but that's mostly a side-show.
HTTPS has at least the following drawbacks:
- it's slower on both sides (but not by much, and possibly hidden by bandwidth constraints)
- it's got higher latency (more round trips, but not many more), this can be minimised with SPDY
- it complicates legitimate analysis and troubleshooting (e.g. packet capture)
- it may prevent some users from accessing your site (corporate policy, content scanning etc.)
- the cert and CA chain can really add to connection overheads (there are potential workarounds like thisthis)
- you have do deal with a CA, and have an administrative process to deal with renewals (sadly more difficult in practice than it sounds)
- more protocols means more configuration, more software, and a greater attack surface
(A less tangible drawback is getting on board the whole PKI/CA train, but let's not go there today...)