Yes and No

It all depends on which city and what airport you're taking. Usually the airports are outside of the city and the train station are central. For example a flight with ryanair from london to dublin in January would cost you about 5£ which is cheaper then any train. However you have to consider the cost of going to the airport.

So if i was you and had a tight budget like $3K for 15 days i would check the prices of a europass (i doubt it would be worth it if you want to stay around the GB), the price of individual train fares and the price of flying + going to the airports.

You can obviously mix it up.