Oral thrush is an upset of the bacterial contents of your mouth- it is usually caused by taking anti-biotics which kill all bacteria- good and bad. You can also get it from taking steroid inhalers for asthma. If you get it, you have white patches in your mouth, your throat is very very dry (no saliva) and you may feel that your breath smells. You may also have a coating on your tongue. For this you take drops, usually.
Tonsillitis is an infection of the tonsils. They act as a gateway to your throat, so are very sensitive to incoming infection. When they suspect infection, they over-react and produce white cells to help fight the infection. Many people have over-zealous tonsils, that over-react and hence get inflamed at the slightest thing. Having tonsillitis all the time is grim and that's why people have them taken out. But the are only trying to do their job!
This is what you have- white blobs on the tonsils. The blobs will come out if prodded but I suggest you leave them alone- they are just trying to fight the infection you have with your cold. You may also feel like your throat is sore. It will pass within a few days but best to stay off work so you don't spread your infection around.