Change in conditions of an ecosystem lead to change in organisms
I will try to provide a general answer, starting point and overview to this question, even though it is rather wide ranging and vague.

The biological process

If we start with your description of the process; you are correct that such a dramatic change in the environment as slash-and-burn agriculture will drastically change the living environments on the canopy floor (well, in the entire ecosystem).

To name a few, there will be changes in:


Light conditions, which will alter the competitive balance between plant species, where ones adapted to a closed canopy will disappear.
Moisture
Temperature (maximum, minimum and variance in temperature) 
Nutrients The point of slash-and-burn agriculture is to provide a boost of nutrients for farming. However, the boost is short-lived and transient. Afterwards, you are usually left with a very nutrient poor soil, which is common in tropical forests. Tropical forests (where slash-and-burn is often used) are high in biomass, but most if it is bound in the standing biomass. When this is removed by clear-cutting, the natural cycling of nutrients is disrupted, and excess nutrients are leached out of the soil (Ricklefs, 2001). 
Loss of host species This will lead to loss of primary consumers (specialists &amp; generalists) feeding on them, and cascading effects into the ecosystem that depends on these.  


All these changes and several others will dramatically modify the niche spaces and the competition between species in the forest patch, and most previous species will probably not be able to persist locally. New species will move in to fill the open niches, and over a (long) period of succession a new quasi-stable community will emerge. Due to the extremely high biodiversity in tropical forests it will almost certainly contain fewer species than the original community.

The question of whether the community will eventually return to its previous state (even if that will take an extremely long time), is a relatively open question, and also depends on the conditions at the specific site and if the spatial scale of the disturbance. The biggest issue for conservation biology is that these land-use changes generally happen at a large scale, so fragmentation of species habitats can lead to regional and even global extinction. Since many tropical species have small ranges and are not adapted to fragmented landscapes, many run the risk of going extinct. Therefore, even if the overall appearance of the original forest will return, it will contain a very different set of species. Land-use changes, even in the form of temporary slash-and-burn agriculture, also generally leads to continued/secondary exploitation of the area and seldom to restoration/regrowth (Meffe &amp; Carroll, 1997).

I also want to point out that you are indirectly referring ("...new equilibrium be reached...") to the idea of a Balance of nature in ecosystems, which is largely seen as obsolete and incorrect. However, as shorthand or when referring to a dynamical equilibrium between multiple populations (with ongoing evolutionary change) it can be meaningful.

The philosophical issue

The question of whether an ecosystem change or extinction of species is "...a bad thing" is inherently philosophical and depends on human judgement. The philosophical foundations for conservation biology can be divided into instrumental values and intrinsic/inherent values. Instrumental values can then be subcategorized as services (pollination, water regulation), goods/resources, information (e.g. genetical info to be used in medical research), and esthetic/spiritual values. Intrinsic values refers to a specific value placed on species, ecosystems or evolutionary history due to its uniqueness and it is seen as independent of human needs and wants (although some argue that values independent of humans are impossible). Intrinsic values are also connected to the acknowledgement that organisms have interests and wants themselves (their own 'good'). Different people naturally weight these issues in different ways, and for some the short-term economical gains will weigh heavier. The long-term economic costs are potentially large, but less certain. From a strict biodiversity perspective (i.e. a species counting perspective), the change will certainly be negative, since the change in land-use will decrease diversity and probably lead to more 'trivial' species moving in. From the perspective of the original ecosystem (if it has a say) the change is per definition negative, since it will disappear. 

The direct intrumental value of individual species are usually uncertain (irrespectivly of the direct rationales used), so the precautionary principle is often invoked to justify species conservation. Also, numerous examples exist of how aspects of biodiversity are related to human values (some recent examples are Garibaldi et al. 2013 and Gamfeldt et al. 2013), such as production or services, which can also be used to justify both conserving biodiversity or individual species. An example of ecosystem services from tropical environements can be seen in Ricketts (2004). However, sometimes functional diversity is emphasized which focus on preserving functional groups of species instead of 'general' biodiversity (i.e. specific species).