I think there are two related issues here. One is the question of code reuse, which many of the other responders have commented on already. In general, if you can easily reuse the code, you should.
The second question is whether it is sensible to have two applications with substantially different behaviors share a common user interface. This is the more interesting question I think. Unfortunately, the answer depends on what your applications are used for. For example, the various applications in an office suite often share a substantial amount of the user interface -- they have common menus, toolbars, dialogs, etc. That's generally considered good, as it makes it easy for a user familiar with one application to work with another from the same suite.
On the other hand, you can imagine cases where sharing the UI will only lead to frustration and agony for your users. As an extreme example, suppose you have an application, SmartHouse 2009, which allows you to control various aspects of your ultra modern smart home. In this application, you have a control that allows you to turn off the cooling system in the house. This involves a button with a clever icon of a melting ice cube or something; when you click the button, a dialog is shown asking the user to confirm the action.
Now suppose you need to write NuclearPowerPlant 2010. Thinking to save some development time, you crib the UI from SmartHouse 2009. Lots of things line up just perfectly. There's even a need for a button to turn off the cooling system in this new application, so you reuse the UI from the first app. Unfortunately, your users have all been using SmartHouse 2009 for months and months, and they have gotten very used to turning off the home A/C with that button. It's ingrained into them that using that button is a relatively harmless action. One day, some poor user comes to the office and logs into NuclearPowerPlant 2010. He feels a little bit cold, so he clicks on the melting ice cube button, expecting it to turn off the A/C. He even clicks through the next dialog -- remember, he's been doing this for months at home. Too bad for him, he disables the power plant cooling system, triggering a massive nuclear explosion.
Obviously this is an extreme fictional example. But you could easily imagine situations in ordinary software, where a user gets used to a button in app A doing a particular kind of action, and assumes that a similar button in app B will do a similar kind of action. If it doesn't, your users will curse you every time they hit that button.
So the question is, how similar are your applications really? Are the conceptually the same, but operating on different types of data, or data from different sources? Are they related in that they are part of a suite of tools with many common actions between the two? Or are they really independent, unrelated applications? If they are unrelated, what is the probability that a user of app A will ever need to use app B?