there are a number of problems to consider with sliders which you won't encounter using text boxes or the checkbox type interactions you also show in your UI snapshot in the question (I won't concern myself with those here, except to say I think they are an excellent faceted search tool).
1. user feedback
it's more tricky to give user feedback in an automated way as the slider is updated. Do you do this each time the user moves a slider, risking clogging up your interface with processing, or let them update with a button once they have selected their values?
2. result counts
As you demonstrate with the checkbox design it's very easy to give the user a count of results that match a set as the page is loaded whereas with sliders (and also with text boxes) this value cannot be set on load meaning the user can end up with empty result sets. However, with text boxes updating via Ajax when they are filled the user is not triggering as much processing as with a slider so you have more time to calculate a result count.
Users arriving at an empty search page should be avoided at all costs.
3. undoing mistakes
Once you have got to a stage where you have restricted the value range too much with a slider it is then cumbersome to undo this when compared to a text field. In a restricted space, such as a website side panel or smart phone, with slider handles too close together further mistakes can easily be made. The initial mistake can be more easily dealt with in the same restricted space when text boxes are used.
4. eye tracking
In the above example the user has to focus on four points at any given time as oppose to only having to focus on two with a pair of text boxes. They are focusing on the slider handles and on the values returned.
This can be helped somewhat by attaching the the values to the handles, but you still have extra brain processing required when compared to simple text boxes. A range is read from left to right (for English speakers as your UI and question implies) and could do without the awkward central interuption of focus.
5. speed of defining a value
Say I want to define the range £100-£200. With text boxes I just type, with sliders I need to first look at the range limits, then slide to investigate what the steps are, then guess where £100 and £200 are, then re-adjust and so on until I am at the place I want to be
6. mouse, keyboard and finger
Sliders afford mouse interaction. They have handles and people are used to using them by clicking and dragging, which is a more cumbersome interaction than focusing and typing.
Text boxes can be focused with both methods and the user is very familar with both mouse and keyboard interactions.
On mobile we use our fingers and the slider is again more cumbersome. Text boxes are a tap to focus and some taps to type, whereas as the slider is a tap to grab and slide to move, compounding point 3 above.
some further reading and sources for my answer:
from this second article, the quote: "Note that boring user interfaces usually work extremely well, because they are intuitive and easy to use" sums up a good position on sliders vs basic text boxes and a maxim that I think should always be considered when weighing up a jquery UI widget against a well established if less technically interesting basic form element.
PS, whatever you decide, use tab index and consider the keyboard focus throughout the interaction