I don't think the distinction has been completely neutralised either. It is certainly the case that in usage these two adverbs are often confused and can sometimes be used with the same meaning.

specially - for a particular purpose

However, when specially is used to mean for a particular purpose, this form of the adverb is the norm:

• This shower gel is specially designed for people with sensitive skins.
  
• This computer programme is specially for children with learning difficulties.
  
• My father made this model aeroplane specially for me.

especially - particularly / above all

We tend to use especially for emphasis, meaning particularly or above all:

• These butterflies are particularly noticeable in April and May, especially in these meadows.
  
• You'll enjoy playing tennis at our local club, especially on weekdays when it's not so busy.

Before adjectives, meaning particularly, especially is more usual:

• The road between Cairo and Alexandria is especially dangerous at night.
  
• It is a bit nippy, but it's not especially cold for this time of year.

special - especial

Note that the adjective especial is rarely used nowadays. Its use is confined to particular contexts where it collocates with particular nouns, e.g. especial interest, especial value when we want to emphasise the exceptional nature of this interest or value:

• The police took especial interest in his activities and watched the house continuously.
  
• The Koh-i-noor diamond, now among the British crown jewels, has especial value as its history dates back to the 14th Century.

In all other cases and contexts, when it means important or different from normal, special is preferred:

• You're a very special person in my life - never forget that.
  
• On special occasions we have wine with our meal, but certainly not every day.
  
• In special cases, prisoners are allowed out on day release twice a week.
  
• He has such ability, I think he'll be the next special adviser to the President.
  
• The special effects in the Lord of the Rings films are quite mind-blowing.
  
• The grapes at the supermarket are on special offer - less than half price.
  

Continual - continuous

Both adjectival forms, continual and continuous, mean without stopping or without a break. They are often used interchangeably:

• This refectory has been in continual /continuous use since the 15th Century.
  
• The continual / continuous croaking of the frogs prevented any sleep that night.

In certain contexts only continuous is possible because continual here would imply that breaks are possible. In these examples, there are clearly no breaks, so continuous is preferred:

• A continuous line of people stretched as far as the eye could see.
  
• They executed the dance in one continuous movement.
  
• The progress of pupils was measured though continuous assessment and not through examinations

When we want to describe things that happen repeatedly, continual is preferred:

• His continual drinking was bound to lead to liver failure one day.
  
• He refused to give up despite the continual warnings of his family.

continually - continuously

The adverbial forms, continually and continuously, are often interchangeable.

• She sniffed continually / continuously all the way through the film and disturbed everyone around her.

But when the meaning is clearly very often, rather than without a break, continually is preferred:

• I've got a very bad stomach upset and I'm continually running to the loo.

Here, continually is behaving as an adverb of frequency, cf. always, all the time, constantly. If we arranged such adverbs along a continuum of frequency, starting with least often and ending with most often, it would read:

• never > rarely > occasionally > sometimes > often >generally > nearly always > constantly/continually