I work with a company that has several web applications available to its customers via HTTPS. Recently, client browsers will not access these web applications due to the HTTPS connection being made via SSLv3 when TLS 1.0 was disabled during an of audit. I understand SSLv3 deprecation is primarily in response to POODLE (SSL3 "POODLE" Vulnerability) as of June 2015 (https://tools.ietf.org/html/rfc7568). TLS 1.0 is not officially deprecated, but seems to be discouraged (e.g. by NIST for the US government, see http://www.nist.gov/itl/csd/tls-043014.cfm, and also at this question Should I disable TLS 1.0 on my servers?).
For these particular web applications, the data is not sensitive to the user or the company, so data confidentiality is not a big issue. However, are there other risks? For example, if users degrade their browser security to accept TLS 1.0, are they at greater risk from a man-in-the-middle attack from other sources? Alternatively, is the company at greater risk from having its server impersonated as part of a man-in-the-middle attack? From a layman's or manager's perspective, what are the risks to the user or company from continuing to use TLS 1.0?
EDIT: The specific cipher suite used is TLS 1.0 with RSA server key for asymmetric exchange and AES 128 bit for the session key. MAC is via SHA-1.