老实 is often used to describe a person being honest, whereas 诚恳 is often used to express a sincere attitude.

老实，是一种较为口语化的表达，通常我们形容某个人的性格时，会用 老实。 而 诚恳 表示 真诚、不虚伪、没有恶意，较为书面。 但实际上也没有太大的差别。（When we talk about one person, we may describe him as brave，patient， 老实 ，etc. When someone is asking you for help，if you want me to give some suggestions whether to help him or not， I may tell you ： His attitude is 诚恳， just help him）

Jeff is right in this case in terms of English expression.

I do not think that 老实 and 诚恳 has any same meanings,indeed. 老实 is one kind of good personality. For instance, if I say 'Tom这个人老实', I mean Tom is honest; doesn't lie to other people; unable to distinguish the bad, the false from the good and the real; always willing to trust other people; more likely to be fooled. and as of 诚恳, this is a word that describes one's words, suggestions, attitude and so on, true-hearted; cordiality; honestly and sincerely.

老实 = honest 诚恳 = in earnest