Long story short: do what @untitledprogrammer said, try both models and cross-validate to help pick one.
Both decision trees (depending on the implementation, e.g. C4.5) and logistic regression should be able to handle continuous and categorical data just fine. For logistic regression, you'll want to dummy code your categorical variables.
As @untitledprogrammer mentioned, it's difficult to know a priori which technique will be better based simply on the types of features you have, continuous or otherwise. It really depends on your specific problem and the data you have. (See No Free Lunch Theorem)
You'll want to keep in mind though that a logistic regression model is searching for a single linear decision boundary in your feature space, whereas a decision tree is essentially partitioning your feature space into half-spaces using axis-aligned linear decision boundaries. The net effect is that you have a non-linear decision boundary, possibly more than one.
This is nice when your data points aren't easily separated by a single hyperplane, but on the other hand, decisions trees are so flexible that they can be prone to overfitting. To combat this, you can try pruning. Logistic regression tends to be less susceptible (but not immune!) to overfitting.
Lastly, another thing to consider is that decision trees can automatically take into account interactions between variables, e.g. $xy$ if you have two independent features $x$ and $y$. With logistic regression, you'll have to manually add those interaction terms yourself.
So you have to ask yourself:
- what kind of decision boundary makes more sense in your particular problem?
- how do you want to balance bias and variance?
- are there interactions between my features?
Of course, it's always a good idea to just try both models and do cross-validation. This will help you find out which one is more likely to have better generalization error.